:: LOPBAN_1 semantic presentation

begin

definition
let X be ( ( ) ( ) set ) ;
let Y be ( ( non empty ) ( non empty ) set ) ;
let F be ( ( Function-like quasi_total ) ( non empty Relation-like [:REAL : ( ( ) ( non empty V34() V144() V145() V146() V150() ) set ) ,Y : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) -defined Y : ( ( non empty ) ( non empty ) set ) -valued Function-like total quasi_total ) Function of [:REAL : ( ( ) ( non empty V34() V144() V145() V146() V150() ) set ) ,Y : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) ,Y : ( ( non empty ) ( non empty ) set ) ) ;
let a be ( ( real ) ( ext-real V31() real ) number ) ;
let f be ( ( Function-like quasi_total ) ( Relation-like X : ( ( ) ( ) set ) -defined Y : ( ( non empty ) ( non empty ) set ) -valued Function-like total quasi_total ) Function of X : ( ( ) ( ) set ) ,Y : ( ( non empty ) ( non empty ) set ) ) ;
:: original: [;]
redefine func F [;] (a,f) -> ( ( ) ( Relation-like X : ( ( ) ( ) AlgebraStr ) -defined Y : ( ( Function-like quasi_total ) ( Relation-like [:X : ( ( ) ( ) AlgebraStr ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) -defined X : ( ( ) ( ) AlgebraStr ) -valued Function-like quasi_total ) Element of bool [:[:X : ( ( ) ( ) AlgebraStr ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like total quasi_total ) Element of Funcs (X : ( ( ) ( ) AlgebraStr ) ,Y : ( ( Function-like quasi_total ) ( Relation-like [:X : ( ( ) ( ) AlgebraStr ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) -defined X : ( ( ) ( ) AlgebraStr ) -valued Function-like quasi_total ) Element of bool [:[:X : ( ( ) ( ) AlgebraStr ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of X : ( ( ) ( ) AlgebraStr ) ,Y : ( ( Function-like quasi_total ) ( Relation-like [:X : ( ( ) ( ) AlgebraStr ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) -defined X : ( ( ) ( ) AlgebraStr ) -valued Function-like quasi_total ) Element of bool [:[:X : ( ( ) ( ) AlgebraStr ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ) ;
end;

definition
let X be ( ( non empty ) ( non empty ) set ) ;
let Y be ( ( non empty ) ( non empty ) addLoopStr ) ;
func FuncAdd (X,Y) -> ( ( Function-like quasi_total ) ( non empty Relation-like [:(Funcs (X : ( ( ) ( ) AlgebraStr ) , the carrier of Y : ( ( Function-like quasi_total ) ( Relation-like [:X : ( ( ) ( ) AlgebraStr ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) -defined X : ( ( ) ( ) AlgebraStr ) -valued Function-like quasi_total ) Element of bool [:[:X : ( ( ) ( ) AlgebraStr ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) )) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of X : ( ( ) ( ) AlgebraStr ) , the carrier of Y : ( ( Function-like quasi_total ) ( Relation-like [:X : ( ( ) ( ) AlgebraStr ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) -defined X : ( ( ) ( ) AlgebraStr ) -valued Function-like quasi_total ) Element of bool [:[:X : ( ( ) ( ) AlgebraStr ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,(Funcs (X : ( ( ) ( ) AlgebraStr ) , the carrier of Y : ( ( Function-like quasi_total ) ( Relation-like [:X : ( ( ) ( ) AlgebraStr ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) -defined X : ( ( ) ( ) AlgebraStr ) -valued Function-like quasi_total ) Element of bool [:[:X : ( ( ) ( ) AlgebraStr ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) )) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of X : ( ( ) ( ) AlgebraStr ) , the carrier of Y : ( ( Function-like quasi_total ) ( Relation-like [:X : ( ( ) ( ) AlgebraStr ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) -defined X : ( ( ) ( ) AlgebraStr ) -valued Function-like quasi_total ) Element of bool [:[:X : ( ( ) ( ) AlgebraStr ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) :] : ( ( ) ( non empty ) set ) -defined Funcs (X : ( ( ) ( ) AlgebraStr ) , the carrier of Y : ( ( Function-like quasi_total ) ( Relation-like [:X : ( ( ) ( ) AlgebraStr ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) -defined X : ( ( ) ( ) AlgebraStr ) -valued Function-like quasi_total ) Element of bool [:[:X : ( ( ) ( ) AlgebraStr ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of X : ( ( ) ( ) AlgebraStr ) , the carrier of Y : ( ( Function-like quasi_total ) ( Relation-like [:X : ( ( ) ( ) AlgebraStr ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) -defined X : ( ( ) ( ) AlgebraStr ) -valued Function-like quasi_total ) Element of bool [:[:X : ( ( ) ( ) AlgebraStr ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like total quasi_total Function-yielding V110() ) BinOp of Funcs (X : ( ( ) ( ) AlgebraStr ) , the carrier of Y : ( ( Function-like quasi_total ) ( Relation-like [:X : ( ( ) ( ) AlgebraStr ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) -defined X : ( ( ) ( ) AlgebraStr ) -valued Function-like quasi_total ) Element of bool [:[:X : ( ( ) ( ) AlgebraStr ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of X : ( ( ) ( ) AlgebraStr ) , the carrier of Y : ( ( Function-like quasi_total ) ( Relation-like [:X : ( ( ) ( ) AlgebraStr ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) -defined X : ( ( ) ( ) AlgebraStr ) -valued Function-like quasi_total ) Element of bool [:[:X : ( ( ) ( ) AlgebraStr ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) means :: LOPBAN_1:def 1
 for f, g being ( ( ) ( Relation-like X : ( ( ) ( ) AlgebraStr ) -defined the carrier of Y : ( ( Function-like quasi_total ) ( Relation-like [:X : ( ( ) ( ) AlgebraStr ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) -defined X : ( ( ) ( ) AlgebraStr ) -valued Function-like quasi_total ) Element of bool [:[:X : ( ( ) ( ) AlgebraStr ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) -valued Function-like total quasi_total ) Element of Funcs (X : ( ( ) ( ) AlgebraStr ) , the carrier of Y : ( ( Function-like quasi_total ) ( Relation-like [:X : ( ( ) ( ) AlgebraStr ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) -defined X : ( ( ) ( ) AlgebraStr ) -valued Function-like quasi_total ) Element of bool [:[:X : ( ( ) ( ) AlgebraStr ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of X : ( ( ) ( ) AlgebraStr ) , the carrier of Y : ( ( Function-like quasi_total ) ( Relation-like [:X : ( ( ) ( ) AlgebraStr ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) -defined X : ( ( ) ( ) AlgebraStr ) -valued Function-like quasi_total ) Element of bool [:[:X : ( ( ) ( ) AlgebraStr ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) holds it : ( ( Function-like quasi_total ) ( Relation-like [:X : ( ( ) ( ) AlgebraStr ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) -defined X : ( ( ) ( ) AlgebraStr ) -valued Function-like quasi_total ) Element of bool [:[:X : ( ( ) ( ) AlgebraStr ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) . (f : ( ( ) ( Relation-like X : ( ( non empty ) ( non empty ) set ) -defined the carrier of Y : ( ( non empty ) ( non empty ) addLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of Funcs (X : ( ( non empty ) ( non empty ) set ) , the carrier of Y : ( ( non empty ) ( non empty ) addLoopStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of X : ( ( non empty ) ( non empty ) set ) , the carrier of Y : ( ( non empty ) ( non empty ) addLoopStr ) : ( ( ) ( non empty ) set ) ) ) ,g : ( ( ) ( Relation-like X : ( ( non empty ) ( non empty ) set ) -defined the carrier of Y : ( ( non empty ) ( non empty ) addLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of Funcs (X : ( ( non empty ) ( non empty ) set ) , the carrier of Y : ( ( non empty ) ( non empty ) addLoopStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of X : ( ( non empty ) ( non empty ) set ) , the carrier of Y : ( ( non empty ) ( non empty ) addLoopStr ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( Relation-like X : ( ( ) ( ) AlgebraStr ) -defined the carrier of Y : ( ( Function-like quasi_total ) ( Relation-like [:X : ( ( ) ( ) AlgebraStr ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) -defined X : ( ( ) ( ) AlgebraStr ) -valued Function-like quasi_total ) Element of bool [:[:X : ( ( ) ( ) AlgebraStr ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) -valued Function-like total quasi_total ) Element of Funcs (X : ( ( ) ( ) AlgebraStr ) , the carrier of Y : ( ( Function-like quasi_total ) ( Relation-like [:X : ( ( ) ( ) AlgebraStr ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) -defined X : ( ( ) ( ) AlgebraStr ) -valued Function-like quasi_total ) Element of bool [:[:X : ( ( ) ( ) AlgebraStr ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of X : ( ( ) ( ) AlgebraStr ) , the carrier of Y : ( ( Function-like quasi_total ) ( Relation-like [:X : ( ( ) ( ) AlgebraStr ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) -defined X : ( ( ) ( ) AlgebraStr ) -valued Function-like quasi_total ) Element of bool [:[:X : ( ( ) ( ) AlgebraStr ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) = the addF of Y : ( ( Function-like quasi_total ) ( Relation-like [:X : ( ( ) ( ) AlgebraStr ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) -defined X : ( ( ) ( ) AlgebraStr ) -valued Function-like quasi_total ) Element of bool [:[:X : ( ( ) ( ) AlgebraStr ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( Relation-like [: the carrier of Y : ( ( Function-like quasi_total ) ( Relation-like [:X : ( ( ) ( ) AlgebraStr ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) -defined X : ( ( ) ( ) AlgebraStr ) -valued Function-like quasi_total ) Element of bool [:[:X : ( ( ) ( ) AlgebraStr ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) , the carrier of Y : ( ( Function-like quasi_total ) ( Relation-like [:X : ( ( ) ( ) AlgebraStr ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) -defined X : ( ( ) ( ) AlgebraStr ) -valued Function-like quasi_total ) Element of bool [:[:X : ( ( ) ( ) AlgebraStr ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) -defined the carrier of Y : ( ( Function-like quasi_total ) ( Relation-like [:X : ( ( ) ( ) AlgebraStr ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) -defined X : ( ( ) ( ) AlgebraStr ) -valued Function-like quasi_total ) Element of bool [:[:X : ( ( ) ( ) AlgebraStr ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) -valued Function-like quasi_total ) Element of bool [:[: the carrier of Y : ( ( Function-like quasi_total ) ( Relation-like [:X : ( ( ) ( ) AlgebraStr ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) -defined X : ( ( ) ( ) AlgebraStr ) -valued Function-like quasi_total ) Element of bool [:[:X : ( ( ) ( ) AlgebraStr ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) , the carrier of Y : ( ( Function-like quasi_total ) ( Relation-like [:X : ( ( ) ( ) AlgebraStr ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) -defined X : ( ( ) ( ) AlgebraStr ) -valued Function-like quasi_total ) Element of bool [:[:X : ( ( ) ( ) AlgebraStr ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) , the carrier of Y : ( ( Function-like quasi_total ) ( Relation-like [:X : ( ( ) ( ) AlgebraStr ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) -defined X : ( ( ) ( ) AlgebraStr ) -valued Function-like quasi_total ) Element of bool [:[:X : ( ( ) ( ) AlgebraStr ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) .: (f : ( ( ) ( Relation-like X : ( ( non empty ) ( non empty ) set ) -defined the carrier of Y : ( ( non empty ) ( non empty ) addLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of Funcs (X : ( ( non empty ) ( non empty ) set ) , the carrier of Y : ( ( non empty ) ( non empty ) addLoopStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of X : ( ( non empty ) ( non empty ) set ) , the carrier of Y : ( ( non empty ) ( non empty ) addLoopStr ) : ( ( ) ( non empty ) set ) ) ) ,g : ( ( ) ( Relation-like X : ( ( non empty ) ( non empty ) set ) -defined the carrier of Y : ( ( non empty ) ( non empty ) addLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of Funcs (X : ( ( non empty ) ( non empty ) set ) , the carrier of Y : ( ( non empty ) ( non empty ) addLoopStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of X : ( ( non empty ) ( non empty ) set ) , the carrier of Y : ( ( non empty ) ( non empty ) addLoopStr ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( Relation-like X : ( ( ) ( ) AlgebraStr ) -defined the carrier of Y : ( ( Function-like quasi_total ) ( Relation-like [:X : ( ( ) ( ) AlgebraStr ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) -defined X : ( ( ) ( ) AlgebraStr ) -valued Function-like quasi_total ) Element of bool [:[:X : ( ( ) ( ) AlgebraStr ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) -valued Function-like total quasi_total ) Element of Funcs (X : ( ( ) ( ) AlgebraStr ) , the carrier of Y : ( ( Function-like quasi_total ) ( Relation-like [:X : ( ( ) ( ) AlgebraStr ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) -defined X : ( ( ) ( ) AlgebraStr ) -valued Function-like quasi_total ) Element of bool [:[:X : ( ( ) ( ) AlgebraStr ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of X : ( ( ) ( ) AlgebraStr ) , the carrier of Y : ( ( Function-like quasi_total ) ( Relation-like [:X : ( ( ) ( ) AlgebraStr ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) -defined X : ( ( ) ( ) AlgebraStr ) -valued Function-like quasi_total ) Element of bool [:[:X : ( ( ) ( ) AlgebraStr ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) ;
end;

definition
let X be ( ( non empty ) ( non empty ) set ) ;
let Y be ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) ;
func FuncExtMult (X,Y) -> ( ( Function-like quasi_total ) ( non empty Relation-like [:REAL : ( ( ) ( non empty V34() V144() V145() V146() V150() ) set ) ,(Funcs (X : ( ( ) ( ) AlgebraStr ) , the carrier of Y : ( ( Function-like quasi_total ) ( Relation-like [:X : ( ( ) ( ) AlgebraStr ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) -defined X : ( ( ) ( ) AlgebraStr ) -valued Function-like quasi_total ) Element of bool [:[:X : ( ( ) ( ) AlgebraStr ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) )) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of X : ( ( ) ( ) AlgebraStr ) , the carrier of Y : ( ( Function-like quasi_total ) ( Relation-like [:X : ( ( ) ( ) AlgebraStr ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) -defined X : ( ( ) ( ) AlgebraStr ) -valued Function-like quasi_total ) Element of bool [:[:X : ( ( ) ( ) AlgebraStr ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) :] : ( ( ) ( non empty ) set ) -defined Funcs (X : ( ( ) ( ) AlgebraStr ) , the carrier of Y : ( ( Function-like quasi_total ) ( Relation-like [:X : ( ( ) ( ) AlgebraStr ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) -defined X : ( ( ) ( ) AlgebraStr ) -valued Function-like quasi_total ) Element of bool [:[:X : ( ( ) ( ) AlgebraStr ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of X : ( ( ) ( ) AlgebraStr ) , the carrier of Y : ( ( Function-like quasi_total ) ( Relation-like [:X : ( ( ) ( ) AlgebraStr ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) -defined X : ( ( ) ( ) AlgebraStr ) -valued Function-like quasi_total ) Element of bool [:[:X : ( ( ) ( ) AlgebraStr ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like total quasi_total Function-yielding V110() ) Function of [:REAL : ( ( ) ( non empty V34() V144() V145() V146() V150() ) set ) ,(Funcs (X : ( ( ) ( ) AlgebraStr ) , the carrier of Y : ( ( Function-like quasi_total ) ( Relation-like [:X : ( ( ) ( ) AlgebraStr ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) -defined X : ( ( ) ( ) AlgebraStr ) -valued Function-like quasi_total ) Element of bool [:[:X : ( ( ) ( ) AlgebraStr ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) )) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of X : ( ( ) ( ) AlgebraStr ) , the carrier of Y : ( ( Function-like quasi_total ) ( Relation-like [:X : ( ( ) ( ) AlgebraStr ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) -defined X : ( ( ) ( ) AlgebraStr ) -valued Function-like quasi_total ) Element of bool [:[:X : ( ( ) ( ) AlgebraStr ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) :] : ( ( ) ( non empty ) set ) , Funcs (X : ( ( ) ( ) AlgebraStr ) , the carrier of Y : ( ( Function-like quasi_total ) ( Relation-like [:X : ( ( ) ( ) AlgebraStr ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) -defined X : ( ( ) ( ) AlgebraStr ) -valued Function-like quasi_total ) Element of bool [:[:X : ( ( ) ( ) AlgebraStr ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of X : ( ( ) ( ) AlgebraStr ) , the carrier of Y : ( ( Function-like quasi_total ) ( Relation-like [:X : ( ( ) ( ) AlgebraStr ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) -defined X : ( ( ) ( ) AlgebraStr ) -valued Function-like quasi_total ) Element of bool [:[:X : ( ( ) ( ) AlgebraStr ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) means :: LOPBAN_1:def 2
 for a being ( ( ) ( ext-real V31() real ) Real)
for f being ( ( ) ( Relation-like X : ( ( ) ( ) AlgebraStr ) -defined the carrier of Y : ( ( Function-like quasi_total ) ( Relation-like [:X : ( ( ) ( ) AlgebraStr ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) -defined X : ( ( ) ( ) AlgebraStr ) -valued Function-like quasi_total ) Element of bool [:[:X : ( ( ) ( ) AlgebraStr ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) -valued Function-like total quasi_total ) Element of Funcs (X : ( ( ) ( ) AlgebraStr ) , the carrier of Y : ( ( Function-like quasi_total ) ( Relation-like [:X : ( ( ) ( ) AlgebraStr ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) -defined X : ( ( ) ( ) AlgebraStr ) -valued Function-like quasi_total ) Element of bool [:[:X : ( ( ) ( ) AlgebraStr ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of X : ( ( ) ( ) AlgebraStr ) , the carrier of Y : ( ( Function-like quasi_total ) ( Relation-like [:X : ( ( ) ( ) AlgebraStr ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) -defined X : ( ( ) ( ) AlgebraStr ) -valued Function-like quasi_total ) Element of bool [:[:X : ( ( ) ( ) AlgebraStr ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) )
for x being ( ( ) ( ) Element of X : ( ( ) ( ) AlgebraStr ) ) holds (it : ( ( Function-like quasi_total ) ( Relation-like [:X : ( ( ) ( ) AlgebraStr ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) -defined X : ( ( ) ( ) AlgebraStr ) -valued Function-like quasi_total ) Element of bool [:[:X : ( ( ) ( ) AlgebraStr ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) . [a : ( ( ) ( ext-real V31() real ) Real) ,f : ( ( ) ( Relation-like X : ( ( non empty ) ( non empty ) set ) -defined the carrier of Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of Funcs (X : ( ( non empty ) ( non empty ) set ) , the carrier of Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of X : ( ( non empty ) ( non empty ) set ) , the carrier of Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) ) ] : ( ( ) ( ) Element of [:REAL : ( ( ) ( non empty V34() V144() V145() V146() V150() ) set ) ,(Funcs (X : ( ( ) ( ) AlgebraStr ) , the carrier of Y : ( ( Function-like quasi_total ) ( Relation-like [:X : ( ( ) ( ) AlgebraStr ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) -defined X : ( ( ) ( ) AlgebraStr ) -valued Function-like quasi_total ) Element of bool [:[:X : ( ( ) ( ) AlgebraStr ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) )) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of X : ( ( ) ( ) AlgebraStr ) , the carrier of Y : ( ( Function-like quasi_total ) ( Relation-like [:X : ( ( ) ( ) AlgebraStr ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) -defined X : ( ( ) ( ) AlgebraStr ) -valued Function-like quasi_total ) Element of bool [:[:X : ( ( ) ( ) AlgebraStr ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) :] : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( Relation-like X : ( ( ) ( ) AlgebraStr ) -defined the carrier of Y : ( ( Function-like quasi_total ) ( Relation-like [:X : ( ( ) ( ) AlgebraStr ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) -defined X : ( ( ) ( ) AlgebraStr ) -valued Function-like quasi_total ) Element of bool [:[:X : ( ( ) ( ) AlgebraStr ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) -valued Function-like total quasi_total ) Element of Funcs (X : ( ( ) ( ) AlgebraStr ) , the carrier of Y : ( ( Function-like quasi_total ) ( Relation-like [:X : ( ( ) ( ) AlgebraStr ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) -defined X : ( ( ) ( ) AlgebraStr ) -valued Function-like quasi_total ) Element of bool [:[:X : ( ( ) ( ) AlgebraStr ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of X : ( ( ) ( ) AlgebraStr ) , the carrier of Y : ( ( Function-like quasi_total ) ( Relation-like [:X : ( ( ) ( ) AlgebraStr ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) -defined X : ( ( ) ( ) AlgebraStr ) -valued Function-like quasi_total ) Element of bool [:[:X : ( ( ) ( ) AlgebraStr ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) . x : ( ( ) ( ) Element of X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of Y : ( ( Function-like quasi_total ) ( Relation-like [:X : ( ( ) ( ) AlgebraStr ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) -defined X : ( ( ) ( ) AlgebraStr ) -valued Function-like quasi_total ) Element of bool [:[:X : ( ( ) ( ) AlgebraStr ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) = a : ( ( ) ( ext-real V31() real ) Real) * (f : ( ( ) ( Relation-like X : ( ( non empty ) ( non empty ) set ) -defined the carrier of Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of Funcs (X : ( ( non empty ) ( non empty ) set ) , the carrier of Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of X : ( ( non empty ) ( non empty ) set ) , the carrier of Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) ) . x : ( ( ) ( ) Element of X : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of Y : ( ( Function-like quasi_total ) ( Relation-like [:X : ( ( ) ( ) AlgebraStr ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) -defined X : ( ( ) ( ) AlgebraStr ) -valued Function-like quasi_total ) Element of bool [:[:X : ( ( ) ( ) AlgebraStr ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) Element of the carrier of Y : ( ( Function-like quasi_total ) ( Relation-like [:X : ( ( ) ( ) AlgebraStr ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) -defined X : ( ( ) ( ) AlgebraStr ) -valued Function-like quasi_total ) Element of bool [:[:X : ( ( ) ( ) AlgebraStr ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ;
end;

definition
let X be ( ( ) ( ) set ) ;
let Y be ( ( non empty ) ( non empty ) ZeroStr ) ;
func FuncZero (X,Y) -> ( ( ) ( Relation-like X : ( ( ) ( ) AlgebraStr ) -defined the carrier of Y : ( ( Function-like quasi_total ) ( Relation-like [:X : ( ( ) ( ) AlgebraStr ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) -defined X : ( ( ) ( ) AlgebraStr ) -valued Function-like quasi_total ) Element of bool [:[:X : ( ( ) ( ) AlgebraStr ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) -valued Function-like total quasi_total ) Element of Funcs (X : ( ( ) ( ) AlgebraStr ) , the carrier of Y : ( ( Function-like quasi_total ) ( Relation-like [:X : ( ( ) ( ) AlgebraStr ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) -defined X : ( ( ) ( ) AlgebraStr ) -valued Function-like quasi_total ) Element of bool [:[:X : ( ( ) ( ) AlgebraStr ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of X : ( ( ) ( ) AlgebraStr ) , the carrier of Y : ( ( Function-like quasi_total ) ( Relation-like [:X : ( ( ) ( ) AlgebraStr ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) -defined X : ( ( ) ( ) AlgebraStr ) -valued Function-like quasi_total ) Element of bool [:[:X : ( ( ) ( ) AlgebraStr ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) equals :: LOPBAN_1:def 3
X : ( ( ) ( ) AlgebraStr ) --> (0. Y : ( ( Function-like quasi_total ) ( Relation-like [:X : ( ( ) ( ) AlgebraStr ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) -defined X : ( ( ) ( ) AlgebraStr ) -valued Function-like quasi_total ) Element of bool [:[:X : ( ( ) ( ) AlgebraStr ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of Y : ( ( Function-like quasi_total ) ( Relation-like [:X : ( ( ) ( ) AlgebraStr ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) -defined X : ( ( ) ( ) AlgebraStr ) -valued Function-like quasi_total ) Element of bool [:[:X : ( ( ) ( ) AlgebraStr ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like quasi_total ) ( Relation-like X : ( ( ) ( ) AlgebraStr ) -defined the carrier of Y : ( ( Function-like quasi_total ) ( Relation-like [:X : ( ( ) ( ) AlgebraStr ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) -defined X : ( ( ) ( ) AlgebraStr ) -valued Function-like quasi_total ) Element of bool [:[:X : ( ( ) ( ) AlgebraStr ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) -valued Function-like constant total quasi_total ) Element of bool [:X : ( ( ) ( ) AlgebraStr ) , the carrier of Y : ( ( Function-like quasi_total ) ( Relation-like [:X : ( ( ) ( ) AlgebraStr ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) -defined X : ( ( ) ( ) AlgebraStr ) -valued Function-like quasi_total ) Element of bool [:[:X : ( ( ) ( ) AlgebraStr ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ;
end;

theorem :: LOPBAN_1:1
for X being ( ( non empty ) ( non empty ) set )
for Y being ( ( non empty ) ( non empty ) addLoopStr )
for f, g, h being ( ( ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty ) ( non empty ) addLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of Funcs (X : ( ( non empty ) ( non empty ) set ) , the carrier of Y : ( ( non empty ) ( non empty ) addLoopStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty ) ( non empty ) addLoopStr ) : ( ( ) ( non empty ) set ) ) ) holds
( h : ( ( ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty ) ( non empty ) addLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty ) ( non empty ) addLoopStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty ) ( non empty ) addLoopStr ) : ( ( ) ( non empty ) set ) ) ) = (FuncAdd (X : ( ( non empty ) ( non empty ) set ) ,Y : ( ( non empty ) ( non empty ) addLoopStr ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like [:(Funcs (b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty ) ( non empty ) addLoopStr ) : ( ( ) ( non empty ) set ) )) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty ) ( non empty ) addLoopStr ) : ( ( ) ( non empty ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty ) ( non empty ) addLoopStr ) : ( ( ) ( non empty ) set ) )) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty ) ( non empty ) addLoopStr ) : ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty ) set ) -defined Funcs (b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty ) ( non empty ) addLoopStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty ) ( non empty ) addLoopStr ) : ( ( ) ( non empty ) set ) ) -valued Function-like total quasi_total Function-yielding V110() ) BinOp of Funcs (b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty ) ( non empty ) addLoopStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty ) ( non empty ) addLoopStr ) : ( ( ) ( non empty ) set ) ) ) . (f : ( ( ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty ) ( non empty ) addLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty ) ( non empty ) addLoopStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty ) ( non empty ) addLoopStr ) : ( ( ) ( non empty ) set ) ) ) ,g : ( ( ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty ) ( non empty ) addLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty ) ( non empty ) addLoopStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty ) ( non empty ) addLoopStr ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty ) ( non empty ) addLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty ) ( non empty ) addLoopStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty ) ( non empty ) addLoopStr ) : ( ( ) ( non empty ) set ) ) ) iff for x being ( ( ) ( ) Element of X : ( ( non empty ) ( non empty ) set ) ) holds h : ( ( ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty ) ( non empty ) addLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty ) ( non empty ) addLoopStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty ) ( non empty ) addLoopStr ) : ( ( ) ( non empty ) set ) ) ) . x : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of b2 : ( ( non empty ) ( non empty ) addLoopStr ) : ( ( ) ( non empty ) set ) ) = (f : ( ( ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty ) ( non empty ) addLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty ) ( non empty ) addLoopStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty ) ( non empty ) addLoopStr ) : ( ( ) ( non empty ) set ) ) ) . x : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of b2 : ( ( non empty ) ( non empty ) addLoopStr ) : ( ( ) ( non empty ) set ) ) + (g : ( ( ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty ) ( non empty ) addLoopStr ) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty ) ( non empty ) addLoopStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty ) ( non empty ) addLoopStr ) : ( ( ) ( non empty ) set ) ) ) . x : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of b2 : ( ( non empty ) ( non empty ) addLoopStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of b2 : ( ( non empty ) ( non empty ) addLoopStr ) : ( ( ) ( non empty ) set ) ) ) ;

theorem :: LOPBAN_1:2
for X being ( ( non empty ) ( non empty ) set )
for Y being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace)
for h, f being ( ( ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of Funcs (X : ( ( non empty ) ( non empty ) set ) , the carrier of Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) )
for a being ( ( ) ( ext-real V31() real ) Real) holds
( h : ( ( ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) ) = (FuncExtMult (X : ( ( non empty ) ( non empty ) set ) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) )) : ( ( Function-like quasi_total ) ( non empty Relation-like [:REAL : ( ( ) ( non empty V34() V144() V145() V146() V150() ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) )) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty ) set ) -defined Funcs (b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) -valued Function-like total quasi_total Function-yielding V110() ) Function of [:REAL : ( ( ) ( non empty V34() V144() V145() V146() V150() ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) )) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty ) set ) , Funcs (b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) ) . [a : ( ( ) ( ext-real V31() real ) Real) ,f : ( ( ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) ) ] : ( ( ) ( ) Element of [:REAL : ( ( ) ( non empty V34() V144() V145() V146() V150() ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) )) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) ) iff for x being ( ( ) ( ) Element of X : ( ( non empty ) ( non empty ) set ) ) holds h : ( ( ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) ) . x : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) = a : ( ( ) ( ext-real V31() real ) Real) * (f : ( ( ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) ) . x : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) ) ;

theorem :: LOPBAN_1:3
for X being ( ( non empty ) ( non empty ) set )
for Y being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace)
for f, g being ( ( ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of Funcs (X : ( ( non empty ) ( non empty ) set ) , the carrier of Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) ) holds (FuncAdd (X : ( ( non empty ) ( non empty ) set ) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) )) : ( ( Function-like quasi_total ) ( non empty Relation-like [:(Funcs (b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) )) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) )) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty ) set ) -defined Funcs (b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) -valued Function-like total quasi_total Function-yielding V110() ) BinOp of Funcs (b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) ) . (f : ( ( ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) ) ,g : ( ( ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) ) = (FuncAdd (X : ( ( non empty ) ( non empty ) set ) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) )) : ( ( Function-like quasi_total ) ( non empty Relation-like [:(Funcs (b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) )) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) )) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty ) set ) -defined Funcs (b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) -valued Function-like total quasi_total Function-yielding V110() ) BinOp of Funcs (b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) ) . (g : ( ( ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) ) ,f : ( ( ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) ) ;

theorem :: LOPBAN_1:4
for X being ( ( non empty ) ( non empty ) set )
for Y being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace)
for f, g, h being ( ( ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of Funcs (X : ( ( non empty ) ( non empty ) set ) , the carrier of Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) ) holds (FuncAdd (X : ( ( non empty ) ( non empty ) set ) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) )) : ( ( Function-like quasi_total ) ( non empty Relation-like [:(Funcs (b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) )) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) )) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty ) set ) -defined Funcs (b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) -valued Function-like total quasi_total Function-yielding V110() ) BinOp of Funcs (b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) ) . (f : ( ( ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) ) ,((FuncAdd (X : ( ( non empty ) ( non empty ) set ) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) )) : ( ( Function-like quasi_total ) ( non empty Relation-like [:(Funcs (b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) )) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) )) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty ) set ) -defined Funcs (b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) -valued Function-like total quasi_total Function-yielding V110() ) BinOp of Funcs (b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) ) . (g : ( ( ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) ) ,h : ( ( ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) ) )) : ( ( ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) ) = (FuncAdd (X : ( ( non empty ) ( non empty ) set ) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) )) : ( ( Function-like quasi_total ) ( non empty Relation-like [:(Funcs (b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) )) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) )) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty ) set ) -defined Funcs (b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) -valued Function-like total quasi_total Function-yielding V110() ) BinOp of Funcs (b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) ) . (((FuncAdd (X : ( ( non empty ) ( non empty ) set ) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) )) : ( ( Function-like quasi_total ) ( non empty Relation-like [:(Funcs (b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) )) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) )) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty ) set ) -defined Funcs (b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) -valued Function-like total quasi_total Function-yielding V110() ) BinOp of Funcs (b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) ) . (f : ( ( ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) ) ,g : ( ( ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) ) )) : ( ( ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) ) ,h : ( ( ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) ) ;

theorem :: LOPBAN_1:5
for X being ( ( non empty ) ( non empty ) set )
for Y being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace)
for f being ( ( ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of Funcs (X : ( ( non empty ) ( non empty ) set ) , the carrier of Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) ) holds (FuncAdd (X : ( ( non empty ) ( non empty ) set ) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) )) : ( ( Function-like quasi_total ) ( non empty Relation-like [:(Funcs (b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) )) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) )) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty ) set ) -defined Funcs (b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) -valued Function-like total quasi_total Function-yielding V110() ) BinOp of Funcs (b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) ) . ((FuncZero (X : ( ( non empty ) ( non empty ) set ) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) )) : ( ( ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) ) ,f : ( ( ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) ) = f : ( ( ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) ) ;

theorem :: LOPBAN_1:6
for X being ( ( non empty ) ( non empty ) set )
for Y being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace)
for f being ( ( ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of Funcs (X : ( ( non empty ) ( non empty ) set ) , the carrier of Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) ) holds (FuncAdd (X : ( ( non empty ) ( non empty ) set ) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) )) : ( ( Function-like quasi_total ) ( non empty Relation-like [:(Funcs (b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) )) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) )) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty ) set ) -defined Funcs (b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) -valued Function-like total quasi_total Function-yielding V110() ) BinOp of Funcs (b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) ) . (f : ( ( ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) ) ,((FuncExtMult (X : ( ( non empty ) ( non empty ) set ) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) )) : ( ( Function-like quasi_total ) ( non empty Relation-like [:REAL : ( ( ) ( non empty V34() V144() V145() V146() V150() ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) )) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty ) set ) -defined Funcs (b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) -valued Function-like total quasi_total Function-yielding V110() ) Function of [:REAL : ( ( ) ( non empty V34() V144() V145() V146() V150() ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) )) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty ) set ) , Funcs (b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) ) . [(- 1 : ( ( ) ( non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural V31() real V113() V114() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V144() V145() V146() V147() V148() V149() V150() ) Element of bool REAL : ( ( ) ( non empty V34() V144() V145() V146() V150() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( ext-real non positive V31() real ) Element of REAL : ( ( ) ( non empty V34() V144() V145() V146() V150() ) set ) ) ,f : ( ( ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) ) ] : ( ( ) ( ) Element of [:REAL : ( ( ) ( non empty V34() V144() V145() V146() V150() ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) )) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) ) = FuncZero (X : ( ( non empty ) ( non empty ) set ) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) ) : ( ( ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) ) ;

theorem :: LOPBAN_1:7
for X being ( ( non empty ) ( non empty ) set )
for Y being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace)
for f being ( ( ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of Funcs (X : ( ( non empty ) ( non empty ) set ) , the carrier of Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) ) holds (FuncExtMult (X : ( ( non empty ) ( non empty ) set ) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) )) : ( ( Function-like quasi_total ) ( non empty Relation-like [:REAL : ( ( ) ( non empty V34() V144() V145() V146() V150() ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) )) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty ) set ) -defined Funcs (b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) -valued Function-like total quasi_total Function-yielding V110() ) Function of [:REAL : ( ( ) ( non empty V34() V144() V145() V146() V150() ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) )) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty ) set ) , Funcs (b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) ) . [1 : ( ( ) ( non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural V31() real V113() V114() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V144() V145() V146() V147() V148() V149() V150() ) Element of bool REAL : ( ( ) ( non empty V34() V144() V145() V146() V150() ) set ) : ( ( ) ( non empty ) set ) ) ) ,f : ( ( ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) ) ] : ( ( ) ( ) Element of [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V144() V145() V146() V147() V148() V149() V150() ) Element of bool REAL : ( ( ) ( non empty V34() V144() V145() V146() V150() ) set ) : ( ( ) ( non empty ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) )) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like Function-like ) set ) = f : ( ( ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) ) ;

theorem :: LOPBAN_1:8
for X being ( ( non empty ) ( non empty ) set )
for Y being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace)
for f being ( ( ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of Funcs (X : ( ( non empty ) ( non empty ) set ) , the carrier of Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) )
for a, b being ( ( ) ( ext-real V31() real ) Real) holds (FuncExtMult (X : ( ( non empty ) ( non empty ) set ) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) )) : ( ( Function-like quasi_total ) ( non empty Relation-like [:REAL : ( ( ) ( non empty V34() V144() V145() V146() V150() ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) )) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty ) set ) -defined Funcs (b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) -valued Function-like total quasi_total Function-yielding V110() ) Function of [:REAL : ( ( ) ( non empty V34() V144() V145() V146() V150() ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) )) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty ) set ) , Funcs (b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) ) . [a : ( ( ) ( ext-real V31() real ) Real) ,((FuncExtMult (X : ( ( non empty ) ( non empty ) set ) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) )) : ( ( Function-like quasi_total ) ( non empty Relation-like [:REAL : ( ( ) ( non empty V34() V144() V145() V146() V150() ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) )) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty ) set ) -defined Funcs (b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) -valued Function-like total quasi_total Function-yielding V110() ) Function of [:REAL : ( ( ) ( non empty V34() V144() V145() V146() V150() ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) )) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty ) set ) , Funcs (b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) ) . [b : ( ( ) ( ext-real V31() real ) Real) ,f : ( ( ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) ) ] : ( ( ) ( ) Element of [:REAL : ( ( ) ( non empty V34() V144() V145() V146() V150() ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) )) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) ) ] : ( ( ) ( ) Element of [:REAL : ( ( ) ( non empty V34() V144() V145() V146() V150() ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) )) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) ) = (FuncExtMult (X : ( ( non empty ) ( non empty ) set ) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) )) : ( ( Function-like quasi_total ) ( non empty Relation-like [:REAL : ( ( ) ( non empty V34() V144() V145() V146() V150() ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) )) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty ) set ) -defined Funcs (b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) -valued Function-like total quasi_total Function-yielding V110() ) Function of [:REAL : ( ( ) ( non empty V34() V144() V145() V146() V150() ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) )) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty ) set ) , Funcs (b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) ) . [(a : ( ( ) ( ext-real V31() real ) Real) * b : ( ( ) ( ext-real V31() real ) Real) ) : ( ( ) ( ext-real V31() real ) Element of REAL : ( ( ) ( non empty V34() V144() V145() V146() V150() ) set ) ) ,f : ( ( ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) ) ] : ( ( ) ( ) Element of [:REAL : ( ( ) ( non empty V34() V144() V145() V146() V150() ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) )) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) ) ;

theorem :: LOPBAN_1:9
for X being ( ( non empty ) ( non empty ) set )
for Y being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace)
for f being ( ( ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of Funcs (X : ( ( non empty ) ( non empty ) set ) , the carrier of Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) )
for a, b being ( ( ) ( ext-real V31() real ) Real) holds (FuncAdd (X : ( ( non empty ) ( non empty ) set ) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) )) : ( ( Function-like quasi_total ) ( non empty Relation-like [:(Funcs (b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) )) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) )) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty ) set ) -defined Funcs (b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) -valued Function-like total quasi_total Function-yielding V110() ) BinOp of Funcs (b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) ) . (((FuncExtMult (X : ( ( non empty ) ( non empty ) set ) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) )) : ( ( Function-like quasi_total ) ( non empty Relation-like [:REAL : ( ( ) ( non empty V34() V144() V145() V146() V150() ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) )) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty ) set ) -defined Funcs (b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) -valued Function-like total quasi_total Function-yielding V110() ) Function of [:REAL : ( ( ) ( non empty V34() V144() V145() V146() V150() ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) )) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty ) set ) , Funcs (b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) ) . [a : ( ( ) ( ext-real V31() real ) Real) ,f : ( ( ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) ) ] : ( ( ) ( ) Element of [:REAL : ( ( ) ( non empty V34() V144() V145() V146() V150() ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) )) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) ) ,((FuncExtMult (X : ( ( non empty ) ( non empty ) set ) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) )) : ( ( Function-like quasi_total ) ( non empty Relation-like [:REAL : ( ( ) ( non empty V34() V144() V145() V146() V150() ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) )) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty ) set ) -defined Funcs (b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) -valued Function-like total quasi_total Function-yielding V110() ) Function of [:REAL : ( ( ) ( non empty V34() V144() V145() V146() V150() ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) )) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty ) set ) , Funcs (b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) ) . [b : ( ( ) ( ext-real V31() real ) Real) ,f : ( ( ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) ) ] : ( ( ) ( ) Element of [:REAL : ( ( ) ( non empty V34() V144() V145() V146() V150() ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) )) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) ) = (FuncExtMult (X : ( ( non empty ) ( non empty ) set ) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) )) : ( ( Function-like quasi_total ) ( non empty Relation-like [:REAL : ( ( ) ( non empty V34() V144() V145() V146() V150() ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) )) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty ) set ) -defined Funcs (b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) -valued Function-like total quasi_total Function-yielding V110() ) Function of [:REAL : ( ( ) ( non empty V34() V144() V145() V146() V150() ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) )) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty ) set ) , Funcs (b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) ) . [(a : ( ( ) ( ext-real V31() real ) Real) + b : ( ( ) ( ext-real V31() real ) Real) ) : ( ( ) ( ext-real V31() real ) Element of REAL : ( ( ) ( non empty V34() V144() V145() V146() V150() ) set ) ) ,f : ( ( ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) ) ] : ( ( ) ( ) Element of [:REAL : ( ( ) ( non empty V34() V144() V145() V146() V150() ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) )) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) ) ;

theorem :: LOPBAN_1:10
for X being ( ( non empty ) ( non empty ) set )
for Y being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) holds RLSStruct(# (Funcs (X : ( ( non empty ) ( non empty ) set ) , the carrier of Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) )) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) ,(FuncZero (X : ( ( non empty ) ( non empty ) set ) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) )) : ( ( ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) ) ,(FuncAdd (X : ( ( non empty ) ( non empty ) set ) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) )) : ( ( Function-like quasi_total ) ( non empty Relation-like [:(Funcs (b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) )) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) )) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty ) set ) -defined Funcs (b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) -valued Function-like total quasi_total Function-yielding V110() ) BinOp of Funcs (b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) ) ,(FuncExtMult (X : ( ( non empty ) ( non empty ) set ) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) )) : ( ( Function-like quasi_total ) ( non empty Relation-like [:REAL : ( ( ) ( non empty V34() V144() V145() V146() V150() ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) )) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty ) set ) -defined Funcs (b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) -valued Function-like total quasi_total Function-yielding V110() ) Function of [:REAL : ( ( ) ( non empty V34() V144() V145() V146() V150() ) set ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) )) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty ) set ) , Funcs (b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) ) #) : ( ( strict ) ( non empty strict ) RLSStruct ) is ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) ;

definition
let X be ( ( non empty ) ( non empty ) set ) ;
let Y be ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) ;
func RealVectSpace (X,Y) -> ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) equals :: LOPBAN_1:def 4
 RLSStruct(# (Funcs (X : ( ( ) ( ) AlgebraStr ) , the carrier of Y : ( ( Function-like quasi_total ) ( Relation-like [:X : ( ( ) ( ) AlgebraStr ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) -defined X : ( ( ) ( ) AlgebraStr ) -valued Function-like quasi_total ) Element of bool [:[:X : ( ( ) ( ) AlgebraStr ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) )) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of X : ( ( ) ( ) AlgebraStr ) , the carrier of Y : ( ( Function-like quasi_total ) ( Relation-like [:X : ( ( ) ( ) AlgebraStr ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) -defined X : ( ( ) ( ) AlgebraStr ) -valued Function-like quasi_total ) Element of bool [:[:X : ( ( ) ( ) AlgebraStr ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,(FuncZero (X : ( ( ) ( ) AlgebraStr ) ,Y : ( ( Function-like quasi_total ) ( Relation-like [:X : ( ( ) ( ) AlgebraStr ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) -defined X : ( ( ) ( ) AlgebraStr ) -valued Function-like quasi_total ) Element of bool [:[:X : ( ( ) ( ) AlgebraStr ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) )) : ( ( ) ( Relation-like X : ( ( ) ( ) AlgebraStr ) -defined the carrier of Y : ( ( Function-like quasi_total ) ( Relation-like [:X : ( ( ) ( ) AlgebraStr ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) -defined X : ( ( ) ( ) AlgebraStr ) -valued Function-like quasi_total ) Element of bool [:[:X : ( ( ) ( ) AlgebraStr ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) -valued Function-like total quasi_total ) Element of Funcs (X : ( ( ) ( ) AlgebraStr ) , the carrier of Y : ( ( Function-like quasi_total ) ( Relation-like [:X : ( ( ) ( ) AlgebraStr ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) -defined X : ( ( ) ( ) AlgebraStr ) -valued Function-like quasi_total ) Element of bool [:[:X : ( ( ) ( ) AlgebraStr ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of X : ( ( ) ( ) AlgebraStr ) , the carrier of Y : ( ( Function-like quasi_total ) ( Relation-like [:X : ( ( ) ( ) AlgebraStr ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) -defined X : ( ( ) ( ) AlgebraStr ) -valued Function-like quasi_total ) Element of bool [:[:X : ( ( ) ( ) AlgebraStr ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) ,(FuncAdd (X : ( ( ) ( ) AlgebraStr ) ,Y : ( ( Function-like quasi_total ) ( Relation-like [:X : ( ( ) ( ) AlgebraStr ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) -defined X : ( ( ) ( ) AlgebraStr ) -valued Function-like quasi_total ) Element of bool [:[:X : ( ( ) ( ) AlgebraStr ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like [:(Funcs (X : ( ( ) ( ) AlgebraStr ) , the carrier of Y : ( ( Function-like quasi_total ) ( Relation-like [:X : ( ( ) ( ) AlgebraStr ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) -defined X : ( ( ) ( ) AlgebraStr ) -valued Function-like quasi_total ) Element of bool [:[:X : ( ( ) ( ) AlgebraStr ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) )) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of X : ( ( ) ( ) AlgebraStr ) , the carrier of Y : ( ( Function-like quasi_total ) ( Relation-like [:X : ( ( ) ( ) AlgebraStr ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) -defined X : ( ( ) ( ) AlgebraStr ) -valued Function-like quasi_total ) Element of bool [:[:X : ( ( ) ( ) AlgebraStr ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,(Funcs (X : ( ( ) ( ) AlgebraStr ) , the carrier of Y : ( ( Function-like quasi_total ) ( Relation-like [:X : ( ( ) ( ) AlgebraStr ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) -defined X : ( ( ) ( ) AlgebraStr ) -valued Function-like quasi_total ) Element of bool [:[:X : ( ( ) ( ) AlgebraStr ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) )) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of X : ( ( ) ( ) AlgebraStr ) , the carrier of Y : ( ( Function-like quasi_total ) ( Relation-like [:X : ( ( ) ( ) AlgebraStr ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) -defined X : ( ( ) ( ) AlgebraStr ) -valued Function-like quasi_total ) Element of bool [:[:X : ( ( ) ( ) AlgebraStr ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) :] : ( ( ) ( non empty ) set ) -defined Funcs (X : ( ( ) ( ) AlgebraStr ) , the carrier of Y : ( ( Function-like quasi_total ) ( Relation-like [:X : ( ( ) ( ) AlgebraStr ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) -defined X : ( ( ) ( ) AlgebraStr ) -valued Function-like quasi_total ) Element of bool [:[:X : ( ( ) ( ) AlgebraStr ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of X : ( ( ) ( ) AlgebraStr ) , the carrier of Y : ( ( Function-like quasi_total ) ( Relation-like [:X : ( ( ) ( ) AlgebraStr ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) -defined X : ( ( ) ( ) AlgebraStr ) -valued Function-like quasi_total ) Element of bool [:[:X : ( ( ) ( ) AlgebraStr ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like total quasi_total Function-yielding V110() ) BinOp of Funcs (X : ( ( ) ( ) AlgebraStr ) , the carrier of Y : ( ( Function-like quasi_total ) ( Relation-like [:X : ( ( ) ( ) AlgebraStr ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) -defined X : ( ( ) ( ) AlgebraStr ) -valued Function-like quasi_total ) Element of bool [:[:X : ( ( ) ( ) AlgebraStr ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of X : ( ( ) ( ) AlgebraStr ) , the carrier of Y : ( ( Function-like quasi_total ) ( Relation-like [:X : ( ( ) ( ) AlgebraStr ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) -defined X : ( ( ) ( ) AlgebraStr ) -valued Function-like quasi_total ) Element of bool [:[:X : ( ( ) ( ) AlgebraStr ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) ,(FuncExtMult (X : ( ( ) ( ) AlgebraStr ) ,Y : ( ( Function-like quasi_total ) ( Relation-like [:X : ( ( ) ( ) AlgebraStr ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) -defined X : ( ( ) ( ) AlgebraStr ) -valued Function-like quasi_total ) Element of bool [:[:X : ( ( ) ( ) AlgebraStr ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like [:REAL : ( ( ) ( non empty V34() V144() V145() V146() V150() ) set ) ,(Funcs (X : ( ( ) ( ) AlgebraStr ) , the carrier of Y : ( ( Function-like quasi_total ) ( Relation-like [:X : ( ( ) ( ) AlgebraStr ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) -defined X : ( ( ) ( ) AlgebraStr ) -valued Function-like quasi_total ) Element of bool [:[:X : ( ( ) ( ) AlgebraStr ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) )) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of X : ( ( ) ( ) AlgebraStr ) , the carrier of Y : ( ( Function-like quasi_total ) ( Relation-like [:X : ( ( ) ( ) AlgebraStr ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) -defined X : ( ( ) ( ) AlgebraStr ) -valued Function-like quasi_total ) Element of bool [:[:X : ( ( ) ( ) AlgebraStr ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) :] : ( ( ) ( non empty ) set ) -defined Funcs (X : ( ( ) ( ) AlgebraStr ) , the carrier of Y : ( ( Function-like quasi_total ) ( Relation-like [:X : ( ( ) ( ) AlgebraStr ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) -defined X : ( ( ) ( ) AlgebraStr ) -valued Function-like quasi_total ) Element of bool [:[:X : ( ( ) ( ) AlgebraStr ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of X : ( ( ) ( ) AlgebraStr ) , the carrier of Y : ( ( Function-like quasi_total ) ( Relation-like [:X : ( ( ) ( ) AlgebraStr ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) -defined X : ( ( ) ( ) AlgebraStr ) -valued Function-like quasi_total ) Element of bool [:[:X : ( ( ) ( ) AlgebraStr ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like total quasi_total Function-yielding V110() ) Function of [:REAL : ( ( ) ( non empty V34() V144() V145() V146() V150() ) set ) ,(Funcs (X : ( ( ) ( ) AlgebraStr ) , the carrier of Y : ( ( Function-like quasi_total ) ( Relation-like [:X : ( ( ) ( ) AlgebraStr ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) -defined X : ( ( ) ( ) AlgebraStr ) -valued Function-like quasi_total ) Element of bool [:[:X : ( ( ) ( ) AlgebraStr ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) )) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of X : ( ( ) ( ) AlgebraStr ) , the carrier of Y : ( ( Function-like quasi_total ) ( Relation-like [:X : ( ( ) ( ) AlgebraStr ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) -defined X : ( ( ) ( ) AlgebraStr ) -valued Function-like quasi_total ) Element of bool [:[:X : ( ( ) ( ) AlgebraStr ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) :] : ( ( ) ( non empty ) set ) , Funcs (X : ( ( ) ( ) AlgebraStr ) , the carrier of Y : ( ( Function-like quasi_total ) ( Relation-like [:X : ( ( ) ( ) AlgebraStr ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) -defined X : ( ( ) ( ) AlgebraStr ) -valued Function-like quasi_total ) Element of bool [:[:X : ( ( ) ( ) AlgebraStr ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of X : ( ( ) ( ) AlgebraStr ) , the carrier of Y : ( ( Function-like quasi_total ) ( Relation-like [:X : ( ( ) ( ) AlgebraStr ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) -defined X : ( ( ) ( ) AlgebraStr ) -valued Function-like quasi_total ) Element of bool [:[:X : ( ( ) ( ) AlgebraStr ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) ,X : ( ( ) ( ) AlgebraStr ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) #) : ( ( strict ) ( non empty strict ) RLSStruct ) ;
end;

registration
let X be ( ( non empty ) ( non empty ) set ) ;
let Y be ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) ;
cluster RealVectSpace (X : ( ( non empty ) ( non empty ) set ) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RLSStruct ) ) : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) -> non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ;
end;

registration
let X be ( ( non empty ) ( non empty ) set ) ;
let Y be ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) ;
cluster RealVectSpace (X : ( ( non empty ) ( non empty ) set ) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RLSStruct ) ) : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) -> non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital constituted-Functions ;
end;

definition
let X be ( ( non empty ) ( non empty ) set ) ;
let Y be ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) ;
let f be ( ( ) ( Relation-like Function-like ) VECTOR of ( ( ) ( non empty ) set ) ) ;
let x be ( ( ) ( ) Element of X : ( ( non empty ) ( non empty ) set ) ) ;
:: original: .
redefine func f . x -> ( ( ) ( ) VECTOR of ( ( ) ( ) set ) ) ;
end;

theorem :: LOPBAN_1:11
for X being ( ( non empty ) ( non empty ) set )
for Y being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace)
for f, g, h being ( ( ) ( Relation-like Function-like ) VECTOR of ( ( ) ( non empty ) set ) ) holds
( h : ( ( ) ( Relation-like Function-like ) VECTOR of ( ( ) ( non empty ) set ) ) = f : ( ( ) ( Relation-like Function-like ) VECTOR of ( ( ) ( non empty ) set ) ) + g : ( ( ) ( Relation-like Function-like ) VECTOR of ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like Function-like ) Element of the carrier of (RealVectSpace (b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) )) : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() constituted-Functions ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) iff for x being ( ( ) ( ) Element of X : ( ( non empty ) ( non empty ) set ) ) holds h : ( ( ) ( Relation-like Function-like ) VECTOR of ( ( ) ( non empty ) set ) ) . x : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) = (f : ( ( ) ( Relation-like Function-like ) VECTOR of ( ( ) ( non empty ) set ) ) . x : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) + (g : ( ( ) ( Relation-like Function-like ) VECTOR of ( ( ) ( non empty ) set ) ) . x : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) ) ;

theorem :: LOPBAN_1:12
for X being ( ( non empty ) ( non empty ) set )
for Y being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace)
for f, h being ( ( ) ( Relation-like Function-like ) VECTOR of ( ( ) ( non empty ) set ) )
for a being ( ( ) ( ext-real V31() real ) Real) holds
( h : ( ( ) ( Relation-like Function-like ) VECTOR of ( ( ) ( non empty ) set ) ) = a : ( ( ) ( ext-real V31() real ) Real) * f : ( ( ) ( Relation-like Function-like ) VECTOR of ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like Function-like ) Element of the carrier of (RealVectSpace (b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) )) : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() constituted-Functions ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) iff for x being ( ( ) ( ) Element of X : ( ( non empty ) ( non empty ) set ) ) holds h : ( ( ) ( Relation-like Function-like ) VECTOR of ( ( ) ( non empty ) set ) ) . x : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) = a : ( ( ) ( ext-real V31() real ) Real) * (f : ( ( ) ( Relation-like Function-like ) VECTOR of ( ( ) ( non empty ) set ) ) . x : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) ) ;

theorem :: LOPBAN_1:13
for X being ( ( non empty ) ( non empty ) set )
for Y being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) holds 0. (RealVectSpace (X : ( ( non empty ) ( non empty ) set ) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) )) : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() constituted-Functions ) RealLinearSpace) : ( ( ) ( Relation-like Function-like V53( RealVectSpace (b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) ) : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() constituted-Functions ) RealLinearSpace) ) ) Element of the carrier of (RealVectSpace (b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) )) : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() constituted-Functions ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) = X : ( ( non empty ) ( non empty ) set ) --> (0. Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) ) : ( ( ) ( V53(b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) ) ) Element of the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) -valued Function-like constant total quasi_total ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ;

begin

definition
let X, Y be ( ( non empty ) ( non empty ) RLSStruct ) ;
let IT be ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of X : ( ( non empty ) ( non empty ) RLSStruct ) : ( ( ) ( non empty ) set ) -defined the carrier of Y : ( ( non empty ) ( non empty ) RLSStruct ) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) ;
attr IT is homogeneous means :: LOPBAN_1:def 5
 for x being ( ( ) ( ) VECTOR of ( ( ) ( ) set ) )
for r being ( ( ) ( ext-real V31() real ) Real) holds IT : ( ( Function-like quasi_total ) ( Relation-like [:X : ( ( ) ( ) set ) ,X : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) -defined X : ( ( ) ( ) set ) -valued Function-like quasi_total ) Element of bool [:[:X : ( ( ) ( ) set ) ,X : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) ,X : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) . (r : ( ( ) ( ext-real V31() real ) Real) * x : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of X : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) : ( ( ) ( ) Element of the carrier of Y : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) : ( ( ) ( ) set ) ) = r : ( ( ) ( ext-real V31() real ) Real) * (IT : ( ( Function-like quasi_total ) ( Relation-like [:X : ( ( ) ( ) set ) ,X : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) -defined X : ( ( ) ( ) set ) -valued Function-like quasi_total ) Element of bool [:[:X : ( ( ) ( ) set ) ,X : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) ,X : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) . x : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of Y : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) : ( ( ) ( ) set ) ) : ( ( ) ( ) Element of the carrier of Y : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) : ( ( ) ( ) set ) ) ;
end;

registration
let X be ( ( non empty ) ( non empty ) RLSStruct ) ;
let Y be ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) ;
cluster non empty Relation-like the carrier of X : ( ( non empty ) ( non empty ) RLSStruct ) : ( ( ) ( non empty ) set ) -defined the carrier of Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RLSStruct ) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total additive homogeneous for ( ( ) ( ) Element of bool [: the carrier of X : ( ( non empty ) ( non empty ) RLSStruct ) : ( ( ) ( non empty ) set ) , the carrier of Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RLSStruct ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ;
end;

definition
let X, Y be ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) ;
mode LinearOperator of X,Y is ( ( Function-like quasi_total additive homogeneous ) ( non empty Relation-like the carrier of X : ( ( non empty ) ( non empty ) RLSStruct ) : ( ( ) ( non empty ) set ) -defined the carrier of Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RLSStruct ) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total additive homogeneous ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) ;
end;

definition
let X, Y be ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) ;
func LinearOperators (X,Y) -> ( ( ) ( ) Subset of ) means :: LOPBAN_1:def 6
 for x being ( ( ) ( ) set ) holds
( x : ( ( ) ( ) set ) in it : ( ( Function-like quasi_total ) ( Relation-like [:X : ( ( non empty ) ( non empty ) RLSStruct ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) -defined X : ( ( non empty ) ( non empty ) RLSStruct ) -valued Function-like quasi_total ) Element of bool [:[:X : ( ( non empty ) ( non empty ) RLSStruct ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) iff x : ( ( ) ( ) set ) is ( ( Function-like quasi_total additive homogeneous ) ( non empty Relation-like the carrier of X : ( ( non empty ) ( non empty ) RLSStruct ) : ( ( ) ( non empty ) set ) -defined the carrier of Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RLSStruct ) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total additive homogeneous ) LinearOperator of X : ( ( non empty ) ( non empty ) RLSStruct ) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RLSStruct ) ) );
end;

registration
let X, Y be ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) ;
cluster LinearOperators (X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RLSStruct ) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RLSStruct ) ) : ( ( ) ( ) Subset of ) -> non empty functional ;
end;

theorem :: LOPBAN_1:14
for X, Y being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) holds LinearOperators (X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) ) : ( ( ) ( non empty functional ) Subset of ) is linearly-closed ;

theorem :: LOPBAN_1:15
for X, Y being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) holds RLSStruct(# (LinearOperators (X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) )) : ( ( ) ( non empty functional ) Subset of ) ,(Zero_ ((LinearOperators (X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) )) : ( ( ) ( non empty functional ) Subset of ) ,(RealVectSpace ( the carrier of X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) )) : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() constituted-Functions ) RealLinearSpace) )) : ( ( ) ( Relation-like Function-like ) Element of LinearOperators (b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) ,b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) ) : ( ( ) ( non empty functional ) Subset of ) ) ,(Add_ ((LinearOperators (X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) )) : ( ( ) ( non empty functional ) Subset of ) ,(RealVectSpace ( the carrier of X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) )) : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() constituted-Functions ) RealLinearSpace) )) : ( ( Function-like quasi_total ) ( non empty Relation-like [:(LinearOperators (b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) ,b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) )) : ( ( ) ( non empty functional ) Subset of ) ,(LinearOperators (b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) ,b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) )) : ( ( ) ( non empty functional ) Subset of ) :] : ( ( ) ( non empty ) set ) -defined LinearOperators (b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) ,b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) ) : ( ( ) ( non empty functional ) Subset of ) -valued Function-like total quasi_total Function-yielding V110() ) Element of bool [:[:(LinearOperators (b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) ,b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) )) : ( ( ) ( non empty functional ) Subset of ) ,(LinearOperators (b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) ,b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) )) : ( ( ) ( non empty functional ) Subset of ) :] : ( ( ) ( non empty ) set ) ,(LinearOperators (b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) ,b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) )) : ( ( ) ( non empty functional ) Subset of ) :] : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,(Mult_ ((LinearOperators (X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) )) : ( ( ) ( non empty functional ) Subset of ) ,(RealVectSpace ( the carrier of X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) )) : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() constituted-Functions ) RealLinearSpace) )) : ( ( Function-like quasi_total ) ( non empty Relation-like [:REAL : ( ( ) ( non empty V34() V144() V145() V146() V150() ) set ) ,(LinearOperators (b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) ,b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) )) : ( ( ) ( non empty functional ) Subset of ) :] : ( ( ) ( non empty ) set ) -defined LinearOperators (b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) ,b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) ) : ( ( ) ( non empty functional ) Subset of ) -valued Function-like total quasi_total Function-yielding V110() ) Element of bool [:[:REAL : ( ( ) ( non empty V34() V144() V145() V146() V150() ) set ) ,(LinearOperators (b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) ,b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) )) : ( ( ) ( non empty functional ) Subset of ) :] : ( ( ) ( non empty ) set ) ,(LinearOperators (b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) ,b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) )) : ( ( ) ( non empty functional ) Subset of ) :] : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) #) : ( ( strict ) ( non empty strict ) RLSStruct ) is ( ( ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) Subspace of RealVectSpace ( the carrier of X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) ) : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() constituted-Functions ) RealLinearSpace) ) ;

registration
let X, Y be ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) ;
cluster RLSStruct(# (LinearOperators (X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RLSStruct ) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RLSStruct ) )) : ( ( ) ( non empty functional ) Subset of ) ,(Zero_ ((LinearOperators (X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RLSStruct ) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RLSStruct ) )) : ( ( ) ( non empty functional ) Subset of ) ,(RealVectSpace ( the carrier of X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RLSStruct ) : ( ( ) ( non empty ) set ) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RLSStruct ) )) : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() constituted-Functions ) RealLinearSpace) )) : ( ( ) ( Relation-like Function-like ) Element of LinearOperators (X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RLSStruct ) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RLSStruct ) ) : ( ( ) ( non empty functional ) Subset of ) ) ,(Add_ ((LinearOperators (X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RLSStruct ) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RLSStruct ) )) : ( ( ) ( non empty functional ) Subset of ) ,(RealVectSpace ( the carrier of X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RLSStruct ) : ( ( ) ( non empty ) set ) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RLSStruct ) )) : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() constituted-Functions ) RealLinearSpace) )) : ( ( Function-like quasi_total ) ( non empty Relation-like [:(LinearOperators (X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RLSStruct ) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RLSStruct ) )) : ( ( ) ( non empty functional ) Subset of ) ,(LinearOperators (X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RLSStruct ) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RLSStruct ) )) : ( ( ) ( non empty functional ) Subset of ) :] : ( ( ) ( non empty ) set ) -defined LinearOperators (X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RLSStruct ) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RLSStruct ) ) : ( ( ) ( non empty functional ) Subset of ) -valued Function-like total quasi_total Function-yielding V110() ) Element of bool [:[:(LinearOperators (X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RLSStruct ) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RLSStruct ) )) : ( ( ) ( non empty functional ) Subset of ) ,(LinearOperators (X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RLSStruct ) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RLSStruct ) )) : ( ( ) ( non empty functional ) Subset of ) :] : ( ( ) ( non empty ) set ) ,(LinearOperators (X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RLSStruct ) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RLSStruct ) )) : ( ( ) ( non empty functional ) Subset of ) :] : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,(Mult_ ((LinearOperators (X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RLSStruct ) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RLSStruct ) )) : ( ( ) ( non empty functional ) Subset of ) ,(RealVectSpace ( the carrier of X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RLSStruct ) : ( ( ) ( non empty ) set ) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RLSStruct ) )) : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() constituted-Functions ) RealLinearSpace) )) : ( ( Function-like quasi_total ) ( non empty Relation-like [:REAL : ( ( ) ( non empty V34() V144() V145() V146() V150() ) set ) ,(LinearOperators (X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RLSStruct ) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RLSStruct ) )) : ( ( ) ( non empty functional ) Subset of ) :] : ( ( ) ( non empty ) set ) -defined LinearOperators (X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RLSStruct ) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RLSStruct ) ) : ( ( ) ( non empty functional ) Subset of ) -valued Function-like total quasi_total Function-yielding V110() ) Element of bool [:[:REAL : ( ( ) ( non empty V34() V144() V145() V146() V150() ) set ) ,(LinearOperators (X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RLSStruct ) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RLSStruct ) )) : ( ( ) ( non empty functional ) Subset of ) :] : ( ( ) ( non empty ) set ) ,(LinearOperators (X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RLSStruct ) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RLSStruct ) )) : ( ( ) ( non empty functional ) Subset of ) :] : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) #) : ( ( strict ) ( non empty strict ) RLSStruct ) -> right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ;
end;

definition
let X, Y be ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) ;
func R_VectorSpace_of_LinearOperators (X,Y) -> ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) equals :: LOPBAN_1:def 7
 RLSStruct(# (LinearOperators (X : ( ( non empty ) ( non empty ) RLSStruct ) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RLSStruct ) )) : ( ( ) ( ) Subset of ) ,(Zero_ ((LinearOperators (X : ( ( non empty ) ( non empty ) RLSStruct ) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RLSStruct ) )) : ( ( ) ( ) Subset of ) ,(RealVectSpace ( the carrier of X : ( ( non empty ) ( non empty ) RLSStruct ) : ( ( ) ( non empty ) set ) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RLSStruct ) )) : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() constituted-Functions ) RealLinearSpace) )) : ( ( ) ( ) Element of LinearOperators (X : ( ( non empty ) ( non empty ) RLSStruct ) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RLSStruct ) ) : ( ( ) ( ) Subset of ) ) ,(Add_ ((LinearOperators (X : ( ( non empty ) ( non empty ) RLSStruct ) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RLSStruct ) )) : ( ( ) ( ) Subset of ) ,(RealVectSpace ( the carrier of X : ( ( non empty ) ( non empty ) RLSStruct ) : ( ( ) ( non empty ) set ) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RLSStruct ) )) : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() constituted-Functions ) RealLinearSpace) )) : ( ( Function-like quasi_total ) ( Relation-like [:(LinearOperators (X : ( ( non empty ) ( non empty ) RLSStruct ) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RLSStruct ) )) : ( ( ) ( ) Subset of ) ,(LinearOperators (X : ( ( non empty ) ( non empty ) RLSStruct ) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RLSStruct ) )) : ( ( ) ( ) Subset of ) :] : ( ( ) ( ) set ) -defined LinearOperators (X : ( ( non empty ) ( non empty ) RLSStruct ) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RLSStruct ) ) : ( ( ) ( ) Subset of ) -valued Function-like quasi_total ) Element of bool [:[:(LinearOperators (X : ( ( non empty ) ( non empty ) RLSStruct ) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RLSStruct ) )) : ( ( ) ( ) Subset of ) ,(LinearOperators (X : ( ( non empty ) ( non empty ) RLSStruct ) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RLSStruct ) )) : ( ( ) ( ) Subset of ) :] : ( ( ) ( ) set ) ,(LinearOperators (X : ( ( non empty ) ( non empty ) RLSStruct ) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RLSStruct ) )) : ( ( ) ( ) Subset of ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ,(Mult_ ((LinearOperators (X : ( ( non empty ) ( non empty ) RLSStruct ) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RLSStruct ) )) : ( ( ) ( ) Subset of ) ,(RealVectSpace ( the carrier of X : ( ( non empty ) ( non empty ) RLSStruct ) : ( ( ) ( non empty ) set ) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RLSStruct ) )) : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() constituted-Functions ) RealLinearSpace) )) : ( ( Function-like quasi_total ) ( Relation-like [:REAL : ( ( ) ( non empty V34() V144() V145() V146() V150() ) set ) ,(LinearOperators (X : ( ( non empty ) ( non empty ) RLSStruct ) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RLSStruct ) )) : ( ( ) ( ) Subset of ) :] : ( ( ) ( ) set ) -defined LinearOperators (X : ( ( non empty ) ( non empty ) RLSStruct ) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RLSStruct ) ) : ( ( ) ( ) Subset of ) -valued Function-like quasi_total ) Element of bool [:[:REAL : ( ( ) ( non empty V34() V144() V145() V146() V150() ) set ) ,(LinearOperators (X : ( ( non empty ) ( non empty ) RLSStruct ) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RLSStruct ) )) : ( ( ) ( ) Subset of ) :] : ( ( ) ( ) set ) ,(LinearOperators (X : ( ( non empty ) ( non empty ) RLSStruct ) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RLSStruct ) )) : ( ( ) ( ) Subset of ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) #) : ( ( strict ) ( strict ) RLSStruct ) ;
end;

registration
let X, Y be ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) ;
cluster R_VectorSpace_of_LinearOperators (X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RLSStruct ) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RLSStruct ) ) : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) -> non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ;
end;

registration
let X, Y be ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) ;
cluster R_VectorSpace_of_LinearOperators (X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RLSStruct ) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RLSStruct ) ) : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) -> non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital constituted-Functions ;
end;

definition
let X, Y be ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) ;
let f be ( ( ) ( Relation-like Function-like ) Element of ( ( ) ( non empty ) set ) ) ;
let v be ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) ;
:: original: .
redefine func f . v -> ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) ;
end;

theorem :: LOPBAN_1:16
for X, Y being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace)
for f, g, h being ( ( ) ( Relation-like Function-like ) VECTOR of ( ( ) ( non empty ) set ) ) holds
( h : ( ( ) ( Relation-like Function-like ) VECTOR of ( ( ) ( non empty ) set ) ) = f : ( ( ) ( Relation-like Function-like ) VECTOR of ( ( ) ( non empty ) set ) ) + g : ( ( ) ( Relation-like Function-like ) VECTOR of ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like Function-like ) Element of the carrier of (R_VectorSpace_of_LinearOperators (b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) ,b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) )) : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() constituted-Functions ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) iff for x being ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) holds h : ( ( ) ( Relation-like Function-like ) VECTOR of ( ( ) ( non empty ) set ) ) . x : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) = (f : ( ( ) ( Relation-like Function-like ) VECTOR of ( ( ) ( non empty ) set ) ) . x : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) + (g : ( ( ) ( Relation-like Function-like ) VECTOR of ( ( ) ( non empty ) set ) ) . x : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) ) ;

theorem :: LOPBAN_1:17
for X, Y being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace)
for f, h being ( ( ) ( Relation-like Function-like ) VECTOR of ( ( ) ( non empty ) set ) )
for a being ( ( ) ( ext-real V31() real ) Real) holds
( h : ( ( ) ( Relation-like Function-like ) VECTOR of ( ( ) ( non empty ) set ) ) = a : ( ( ) ( ext-real V31() real ) Real) * f : ( ( ) ( Relation-like Function-like ) VECTOR of ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like Function-like ) Element of the carrier of (R_VectorSpace_of_LinearOperators (b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) ,b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) )) : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() constituted-Functions ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) iff for x being ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) holds h : ( ( ) ( Relation-like Function-like ) VECTOR of ( ( ) ( non empty ) set ) ) . x : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) = a : ( ( ) ( ext-real V31() real ) Real) * (f : ( ( ) ( Relation-like Function-like ) VECTOR of ( ( ) ( non empty ) set ) ) . x : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) ) ;

theorem :: LOPBAN_1:18
for X, Y being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) holds 0. (R_VectorSpace_of_LinearOperators (X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) )) : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() constituted-Functions ) RealLinearSpace) : ( ( ) ( Relation-like Function-like V53( R_VectorSpace_of_LinearOperators (b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) ,b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) ) : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() constituted-Functions ) RealLinearSpace) ) ) Element of the carrier of (R_VectorSpace_of_LinearOperators (b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) ,b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) )) : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() constituted-Functions ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) = the carrier of X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) --> (0. Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) ) : ( ( ) ( V53(b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) ) ) Element of the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) -valued Function-like constant total quasi_total ) Element of bool [: the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ;

theorem :: LOPBAN_1:19
for X, Y being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) holds the carrier of X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) --> (0. Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) ) : ( ( ) ( V53(b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) ) ) Element of the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) -valued Function-like constant total quasi_total ) Element of bool [: the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) is ( ( Function-like quasi_total additive homogeneous ) ( non empty Relation-like the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total additive homogeneous ) LinearOperator of X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) ) ;

begin

theorem :: LOPBAN_1:20
for X being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace)
for seq being ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V144() V145() V146() V147() V148() V149() V150() ) Element of bool REAL : ( ( ) ( non empty V34() V144() V145() V146() V150() ) set ) : ( ( ) ( non empty ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) )
for g being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) st seq : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V144() V145() V146() V147() V148() V149() V150() ) Element of bool REAL : ( ( ) ( non empty V34() V144() V145() V146() V150() ) set ) : ( ( ) ( non empty ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) is convergent & lim seq : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V144() V145() V146() V147() V148() V149() V150() ) Element of bool REAL : ( ( ) ( non empty V34() V144() V145() V146() V150() ) set ) : ( ( ) ( non empty ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) = g : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) holds
( ||.seq : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V144() V145() V146() V147() V148() V149() V150() ) Element of bool REAL : ( ( ) ( non empty V34() V144() V145() V146() V150() ) set ) : ( ( ) ( non empty ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) .|| : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V144() V145() V146() V147() V148() V149() V150() ) Element of bool REAL : ( ( ) ( non empty V34() V144() V145() V146() V150() ) set ) : ( ( ) ( non empty ) set ) ) -defined REAL : ( ( ) ( non empty V34() V144() V145() V146() V150() ) set ) -valued Function-like total quasi_total V134() V135() V136() ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V144() V145() V146() V147() V148() V149() V150() ) Element of bool REAL : ( ( ) ( non empty V34() V144() V145() V146() V150() ) set ) : ( ( ) ( non empty ) set ) ) ,REAL : ( ( ) ( non empty V34() V144() V145() V146() V150() ) set ) :] : ( ( ) ( non empty V134() V135() V136() ) set ) : ( ( ) ( non empty ) set ) ) is convergent & lim ||.seq : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V144() V145() V146() V147() V148() V149() V150() ) Element of bool REAL : ( ( ) ( non empty V34() V144() V145() V146() V150() ) set ) : ( ( ) ( non empty ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) .|| : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V144() V145() V146() V147() V148() V149() V150() ) Element of bool REAL : ( ( ) ( non empty V34() V144() V145() V146() V150() ) set ) : ( ( ) ( non empty ) set ) ) -defined REAL : ( ( ) ( non empty V34() V144() V145() V146() V150() ) set ) -valued Function-like total quasi_total V134() V135() V136() ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V144() V145() V146() V147() V148() V149() V150() ) Element of bool REAL : ( ( ) ( non empty V34() V144() V145() V146() V150() ) set ) : ( ( ) ( non empty ) set ) ) ,REAL : ( ( ) ( non empty V34() V144() V145() V146() V150() ) set ) :] : ( ( ) ( non empty V134() V135() V136() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ext-real V31() real ) Element of REAL : ( ( ) ( non empty V34() V144() V145() V146() V150() ) set ) ) = ||.g : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) .|| : ( ( ) ( ext-real non negative V31() real ) Element of REAL : ( ( ) ( non empty V34() V144() V145() V146() V150() ) set ) ) ) ;

definition
let X, Y be ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) ;
let IT be ( ( Function-like quasi_total additive homogeneous ) ( non empty Relation-like the carrier of X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total additive homogeneous ) LinearOperator of X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) ) ;
attr IT is Lipschitzian means :: LOPBAN_1:def 8
 ex K being ( ( ) ( ext-real V31() real ) Real) st
( 0 : ( ( ) ( empty Function-like functional ext-real non positive non negative epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V31() real V113() V114() V144() V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V144() V145() V146() V147() V148() V149() V150() ) Element of bool REAL : ( ( ) ( non empty V34() V144() V145() V146() V150() ) set ) : ( ( ) ( non empty ) set ) ) ) <= K : ( ( ) ( ext-real V31() real ) Real) & ( for x being ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) holds ||.(IT : ( ( Function-like quasi_total ) ( Relation-like [:X : ( ( non empty ) ( non empty ) RLSStruct ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) -defined X : ( ( non empty ) ( non empty ) RLSStruct ) -valued Function-like quasi_total ) Element of bool [:[:X : ( ( non empty ) ( non empty ) RLSStruct ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) ,X : ( ( non empty ) ( non empty ) RLSStruct ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) . x : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RLSStruct ) : ( ( ) ( non empty ) set ) ) .|| : ( ( ) ( ext-real V31() real ) Element of REAL : ( ( ) ( non empty V34() V144() V145() V146() V150() ) set ) ) <= K : ( ( ) ( ext-real V31() real ) Real) * ||.x : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) .|| : ( ( ) ( ext-real V31() real ) Element of REAL : ( ( ) ( non empty V34() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ext-real V31() real ) Element of REAL : ( ( ) ( non empty V34() V144() V145() V146() V150() ) set ) ) ) );
end;

theorem :: LOPBAN_1:21
for X, Y being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace)
for f being ( ( Function-like quasi_total additive homogeneous ) ( non empty Relation-like the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total additive homogeneous ) LinearOperator of X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) ) st ( for x being ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) holds f : ( ( Function-like quasi_total additive homogeneous ) ( non empty Relation-like the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total additive homogeneous ) LinearOperator of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) ,b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) ) . x : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) = 0. Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) : ( ( ) ( V53(b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) ) ) Element of the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) ) holds
f : ( ( Function-like quasi_total additive homogeneous ) ( non empty Relation-like the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total additive homogeneous ) LinearOperator of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) ,b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) ) is Lipschitzian ;

registration
let X, Y be ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) ;
cluster non empty Relation-like the carrier of X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) : ( ( ) ( non empty ) set ) -defined the carrier of Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total additive homogeneous Lipschitzian for ( ( ) ( ) Element of bool [: the carrier of X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) : ( ( ) ( non empty ) set ) , the carrier of Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ;
end;

definition
let X, Y be ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) ;
func BoundedLinearOperators (X,Y) -> ( ( ) ( ) Subset of ) means :: LOPBAN_1:def 9
 for x being ( ( ) ( ) set ) holds
( x : ( ( ) ( ) set ) in it : ( ( Function-like quasi_total ) ( Relation-like [:X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) ,X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) :] : ( ( ) ( ) set ) -defined X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) -valued Function-like quasi_total ) Element of bool [:[:X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) ,X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) :] : ( ( ) ( ) set ) ,X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) iff x : ( ( ) ( ) set ) is ( ( Function-like quasi_total additive homogeneous Lipschitzian ) ( non empty Relation-like the carrier of X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) : ( ( ) ( non empty ) set ) -defined the carrier of Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total additive homogeneous Lipschitzian ) LinearOperator of X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) ) );
end;

registration
let X, Y be ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) ;
cluster BoundedLinearOperators (X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( ) Subset of ) -> non empty ;
end;

theorem :: LOPBAN_1:22
for X, Y being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) holds BoundedLinearOperators (X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) ) : ( ( ) ( non empty ) Subset of ) is linearly-closed ;

theorem :: LOPBAN_1:23
for X, Y being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) holds RLSStruct(# (BoundedLinearOperators (X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) )) : ( ( ) ( non empty ) Subset of ) ,(Zero_ ((BoundedLinearOperators (X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) )) : ( ( ) ( non empty ) Subset of ) ,(R_VectorSpace_of_LinearOperators (X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) )) : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() constituted-Functions ) RealLinearSpace) )) : ( ( ) ( ) Element of BoundedLinearOperators (b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) ,b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) ) : ( ( ) ( non empty ) Subset of ) ) ,(Add_ ((BoundedLinearOperators (X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) )) : ( ( ) ( non empty ) Subset of ) ,(R_VectorSpace_of_LinearOperators (X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) )) : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() constituted-Functions ) RealLinearSpace) )) : ( ( Function-like quasi_total ) ( non empty Relation-like [:(BoundedLinearOperators (b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) ,b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) )) : ( ( ) ( non empty ) Subset of ) ,(BoundedLinearOperators (b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) ,b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) )) : ( ( ) ( non empty ) Subset of ) :] : ( ( ) ( non empty ) set ) -defined BoundedLinearOperators (b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) ,b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) ) : ( ( ) ( non empty ) Subset of ) -valued Function-like total quasi_total ) Element of bool [:[:(BoundedLinearOperators (b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) ,b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) )) : ( ( ) ( non empty ) Subset of ) ,(BoundedLinearOperators (b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) ,b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) )) : ( ( ) ( non empty ) Subset of ) :] : ( ( ) ( non empty ) set ) ,(BoundedLinearOperators (b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) ,b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) )) : ( ( ) ( non empty ) Subset of ) :] : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,(Mult_ ((BoundedLinearOperators (X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) )) : ( ( ) ( non empty ) Subset of ) ,(R_VectorSpace_of_LinearOperators (X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) )) : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() constituted-Functions ) RealLinearSpace) )) : ( ( Function-like quasi_total ) ( non empty Relation-like [:REAL : ( ( ) ( non empty V34() V144() V145() V146() V150() ) set ) ,(BoundedLinearOperators (b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) ,b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) )) : ( ( ) ( non empty ) Subset of ) :] : ( ( ) ( non empty ) set ) -defined BoundedLinearOperators (b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) ,b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) ) : ( ( ) ( non empty ) Subset of ) -valued Function-like total quasi_total ) Element of bool [:[:REAL : ( ( ) ( non empty V34() V144() V145() V146() V150() ) set ) ,(BoundedLinearOperators (b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) ,b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) )) : ( ( ) ( non empty ) Subset of ) :] : ( ( ) ( non empty ) set ) ,(BoundedLinearOperators (b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) ,b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) )) : ( ( ) ( non empty ) Subset of ) :] : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) #) : ( ( strict ) ( non empty strict ) RLSStruct ) is ( ( ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) Subspace of R_VectorSpace_of_LinearOperators (X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) ) : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() constituted-Functions ) RealLinearSpace) ) ;

registration
let X, Y be ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) ;
cluster RLSStruct(# (BoundedLinearOperators (X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) )) : ( ( ) ( non empty ) Subset of ) ,(Zero_ ((BoundedLinearOperators (X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) )) : ( ( ) ( non empty ) Subset of ) ,(R_VectorSpace_of_LinearOperators (X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) )) : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() constituted-Functions ) RealLinearSpace) )) : ( ( ) ( ) Element of BoundedLinearOperators (X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( non empty ) Subset of ) ) ,(Add_ ((BoundedLinearOperators (X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) )) : ( ( ) ( non empty ) Subset of ) ,(R_VectorSpace_of_LinearOperators (X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) )) : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() constituted-Functions ) RealLinearSpace) )) : ( ( Function-like quasi_total ) ( non empty Relation-like [:(BoundedLinearOperators (X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) )) : ( ( ) ( non empty ) Subset of ) ,(BoundedLinearOperators (X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) )) : ( ( ) ( non empty ) Subset of ) :] : ( ( ) ( non empty ) set ) -defined BoundedLinearOperators (X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( non empty ) Subset of ) -valued Function-like total quasi_total ) Element of bool [:[:(BoundedLinearOperators (X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) )) : ( ( ) ( non empty ) Subset of ) ,(BoundedLinearOperators (X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) )) : ( ( ) ( non empty ) Subset of ) :] : ( ( ) ( non empty ) set ) ,(BoundedLinearOperators (X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) )) : ( ( ) ( non empty ) Subset of ) :] : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,(Mult_ ((BoundedLinearOperators (X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) )) : ( ( ) ( non empty ) Subset of ) ,(R_VectorSpace_of_LinearOperators (X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) )) : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() constituted-Functions ) RealLinearSpace) )) : ( ( Function-like quasi_total ) ( non empty Relation-like [:REAL : ( ( ) ( non empty V34() V144() V145() V146() V150() ) set ) ,(BoundedLinearOperators (X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) )) : ( ( ) ( non empty ) Subset of ) :] : ( ( ) ( non empty ) set ) -defined BoundedLinearOperators (X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( non empty ) Subset of ) -valued Function-like total quasi_total ) Element of bool [:[:REAL : ( ( ) ( non empty V34() V144() V145() V146() V150() ) set ) ,(BoundedLinearOperators (X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) )) : ( ( ) ( non empty ) Subset of ) :] : ( ( ) ( non empty ) set ) ,(BoundedLinearOperators (X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) )) : ( ( ) ( non empty ) Subset of ) :] : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) #) : ( ( strict ) ( non empty strict ) RLSStruct ) -> right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ;
end;

definition
let X, Y be ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) ;
func R_VectorSpace_of_BoundedLinearOperators (X,Y) -> ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) equals :: LOPBAN_1:def 10
 RLSStruct(# (BoundedLinearOperators (X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) )) : ( ( ) ( non empty ) Subset of ) ,(Zero_ ((BoundedLinearOperators (X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) )) : ( ( ) ( non empty ) Subset of ) ,(R_VectorSpace_of_LinearOperators (X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) )) : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() constituted-Functions ) RealLinearSpace) )) : ( ( ) ( ) Element of BoundedLinearOperators (X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( non empty ) Subset of ) ) ,(Add_ ((BoundedLinearOperators (X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) )) : ( ( ) ( non empty ) Subset of ) ,(R_VectorSpace_of_LinearOperators (X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) )) : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() constituted-Functions ) RealLinearSpace) )) : ( ( Function-like quasi_total ) ( non empty Relation-like [:(BoundedLinearOperators (X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) )) : ( ( ) ( non empty ) Subset of ) ,(BoundedLinearOperators (X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) )) : ( ( ) ( non empty ) Subset of ) :] : ( ( ) ( non empty ) set ) -defined BoundedLinearOperators (X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( non empty ) Subset of ) -valued Function-like total quasi_total ) Element of bool [:[:(BoundedLinearOperators (X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) )) : ( ( ) ( non empty ) Subset of ) ,(BoundedLinearOperators (X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) )) : ( ( ) ( non empty ) Subset of ) :] : ( ( ) ( non empty ) set ) ,(BoundedLinearOperators (X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) )) : ( ( ) ( non empty ) Subset of ) :] : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,(Mult_ ((BoundedLinearOperators (X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) )) : ( ( ) ( non empty ) Subset of ) ,(R_VectorSpace_of_LinearOperators (X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) )) : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() constituted-Functions ) RealLinearSpace) )) : ( ( Function-like quasi_total ) ( non empty Relation-like [:REAL : ( ( ) ( non empty V34() V144() V145() V146() V150() ) set ) ,(BoundedLinearOperators (X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) )) : ( ( ) ( non empty ) Subset of ) :] : ( ( ) ( non empty ) set ) -defined BoundedLinearOperators (X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( non empty ) Subset of ) -valued Function-like total quasi_total ) Element of bool [:[:REAL : ( ( ) ( non empty V34() V144() V145() V146() V150() ) set ) ,(BoundedLinearOperators (X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) )) : ( ( ) ( non empty ) Subset of ) :] : ( ( ) ( non empty ) set ) ,(BoundedLinearOperators (X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) )) : ( ( ) ( non empty ) Subset of ) :] : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) #) : ( ( strict ) ( non empty left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RLSStruct ) ;
end;

registration
let X, Y be ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) ;
cluster R_VectorSpace_of_BoundedLinearOperators (X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) ) : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) -> non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ;
end;

registration
let X, Y be ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) ;
cluster -> Relation-like Function-like for ( ( ) ( ) Element of the carrier of (R_VectorSpace_of_BoundedLinearOperators (X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) )) : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) ;
end;

definition
let X, Y be ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) ;
let f be ( ( ) ( Relation-like Function-like ) Element of ( ( ) ( non empty ) set ) ) ;
let v be ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) ;
:: original: .
redefine func f . v -> ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) ;
end;

theorem :: LOPBAN_1:24
for X, Y being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace)
for f, g, h being ( ( ) ( Relation-like Function-like ) VECTOR of ( ( ) ( non empty ) set ) ) holds
( h : ( ( ) ( Relation-like Function-like ) VECTOR of ( ( ) ( non empty ) set ) ) = f : ( ( ) ( Relation-like Function-like ) VECTOR of ( ( ) ( non empty ) set ) ) + g : ( ( ) ( Relation-like Function-like ) VECTOR of ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like Function-like ) Element of the carrier of (R_VectorSpace_of_BoundedLinearOperators (b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) ,b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) )) : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) iff for x being ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) holds h : ( ( ) ( Relation-like Function-like ) VECTOR of ( ( ) ( non empty ) set ) ) . x : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) = (f : ( ( ) ( Relation-like Function-like ) VECTOR of ( ( ) ( non empty ) set ) ) . x : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) + (g : ( ( ) ( Relation-like Function-like ) VECTOR of ( ( ) ( non empty ) set ) ) . x : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) ) ;

theorem :: LOPBAN_1:25
for X, Y being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace)
for f, h being ( ( ) ( Relation-like Function-like ) VECTOR of ( ( ) ( non empty ) set ) )
for a being ( ( ) ( ext-real V31() real ) Real) holds
( h : ( ( ) ( Relation-like Function-like ) VECTOR of ( ( ) ( non empty ) set ) ) = a : ( ( ) ( ext-real V31() real ) Real) * f : ( ( ) ( Relation-like Function-like ) VECTOR of ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like Function-like ) Element of the carrier of (R_VectorSpace_of_BoundedLinearOperators (b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) ,b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) )) : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) iff for x being ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) holds h : ( ( ) ( Relation-like Function-like ) VECTOR of ( ( ) ( non empty ) set ) ) . x : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) = a : ( ( ) ( ext-real V31() real ) Real) * (f : ( ( ) ( Relation-like Function-like ) VECTOR of ( ( ) ( non empty ) set ) ) . x : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) ) ;

theorem :: LOPBAN_1:26
for X, Y being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) holds 0. (R_VectorSpace_of_BoundedLinearOperators (X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) )) : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( Relation-like Function-like V53( R_VectorSpace_of_BoundedLinearOperators (b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) ,b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) ) : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) ) ) Element of the carrier of (R_VectorSpace_of_BoundedLinearOperators (b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) ,b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) )) : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() ) RealLinearSpace) : ( ( ) ( non empty ) set ) ) = the carrier of X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) --> (0. Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) ) : ( ( ) ( V53(b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) ) ) Element of the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like constant total quasi_total ) Element of bool [: the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ;

definition
let X, Y be ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) ;
let f be ( ( ) ( ) set ) ;
assume f : ( ( ) ( ) set ) in BoundedLinearOperators (X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) ) : ( ( ) ( non empty ) Subset of ) ;
func modetrans (f,X,Y) -> ( ( Function-like quasi_total additive homogeneous Lipschitzian ) ( non empty Relation-like the carrier of X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) : ( ( ) ( non empty ) set ) -defined the carrier of Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total additive homogeneous Lipschitzian ) LinearOperator of X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) ) equals :: LOPBAN_1:def 11
f : ( ( Function-like quasi_total ) ( Relation-like [:X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) ,X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) :] : ( ( ) ( ) set ) -defined X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) -valued Function-like quasi_total ) Element of bool [:[:X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) ,X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) :] : ( ( ) ( ) set ) ,X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ;
end;

definition
let X, Y be ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) ;
let u be ( ( Function-like quasi_total additive homogeneous ) ( non empty Relation-like the carrier of X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total additive homogeneous ) LinearOperator of X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) ) ;
func PreNorms u -> ( ( non empty ) ( non empty V144() V145() V146() ) Subset of ( ( ) ( non empty ) set ) ) equals :: LOPBAN_1:def 12
 { ||.(u : ( ( Function-like quasi_total ) ( Relation-like [:X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) ,X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) :] : ( ( ) ( ) set ) -defined X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) -valued Function-like quasi_total ) Element of bool [:[:X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) ,X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) :] : ( ( ) ( ) set ) ,X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) . t : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) : ( ( ) ( non empty ) set ) ) .|| : ( ( ) ( ext-real non negative V31() real ) Element of REAL : ( ( ) ( non empty V34() V144() V145() V146() V150() ) set ) ) where t is ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) : ||.t : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) .|| : ( ( ) ( ext-real V31() real ) Element of REAL : ( ( ) ( non empty V34() V144() V145() V146() V150() ) set ) ) <= 1 : ( ( ) ( non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural V31() real V113() V114() V144() V145() V146() V147() V148() V149() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V144() V145() V146() V147() V148() V149() V150() ) Element of bool REAL : ( ( ) ( non empty V34() V144() V145() V146() V150() ) set ) : ( ( ) ( non empty ) set ) ) ) } ;
end;

theorem :: LOPBAN_1:27
for X, Y being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace)
for g being ( ( Function-like quasi_total additive homogeneous Lipschitzian ) ( non empty Relation-like the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total additive homogeneous Lipschitzian ) LinearOperator of X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) ) holds PreNorms g : ( ( Function-like quasi_total additive homogeneous Lipschitzian ) ( non empty Relation-like the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total additive homogeneous Lipschitzian ) LinearOperator of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) ,b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) ) : ( ( non empty ) ( non empty V144() V145() V146() ) Subset of ( ( ) ( non empty ) set ) ) is bounded_above ;

theorem :: LOPBAN_1:28
for X, Y being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace)
for g being ( ( Function-like quasi_total additive homogeneous ) ( non empty Relation-like the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total additive homogeneous ) LinearOperator of X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) ) holds
( g : ( ( Function-like quasi_total additive homogeneous ) ( non empty Relation-like the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total additive homogeneous ) LinearOperator of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) ,b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) ) is Lipschitzian iff PreNorms g : ( ( Function-like quasi_total additive homogeneous ) ( non empty Relation-like the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total additive homogeneous ) LinearOperator of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) ,b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) ) : ( ( non empty ) ( non empty V144() V145() V146() ) Subset of ( ( ) ( non empty ) set ) ) is bounded_above ) ;

definition
let X, Y be ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) ;
func BoundedLinearOperatorsNorm (X,Y) -> ( ( Function-like quasi_total ) ( non empty Relation-like BoundedLinearOperators (X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( non empty ) Subset of ) -defined REAL : ( ( ) ( non empty V34() V144() V145() V146() V150() ) set ) -valued Function-like total quasi_total V134() V135() V136() ) Function of BoundedLinearOperators (X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( non empty ) Subset of ) , REAL : ( ( ) ( non empty V34() V144() V145() V146() V150() ) set ) ) means :: LOPBAN_1:def 13
 for x being ( ( ) ( ) set ) st x : ( ( ) ( ) set ) in BoundedLinearOperators (X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( non empty ) Subset of ) holds
it : ( ( Function-like quasi_total ) ( Relation-like [:X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) ,X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) :] : ( ( ) ( ) set ) -defined X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) -valued Function-like quasi_total ) Element of bool [:[:X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) ,X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) :] : ( ( ) ( ) set ) ,X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) . x : ( ( ) ( ) set ) : ( ( ) ( ext-real V31() real ) Element of REAL : ( ( ) ( non empty V34() V144() V145() V146() V150() ) set ) ) = upper_bound (PreNorms (modetrans (x : ( ( ) ( ) set ) ,X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) )) : ( ( Function-like quasi_total additive homogeneous Lipschitzian ) ( non empty Relation-like the carrier of X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) : ( ( ) ( non empty ) set ) -defined the carrier of Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total additive homogeneous Lipschitzian ) LinearOperator of X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) ) ) : ( ( non empty ) ( non empty V144() V145() V146() ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( ext-real V31() real ) Element of REAL : ( ( ) ( non empty V34() V144() V145() V146() V150() ) set ) ) ;
end;

theorem :: LOPBAN_1:29
for X, Y being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace)
for f being ( ( Function-like quasi_total additive homogeneous Lipschitzian ) ( non empty Relation-like the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total additive homogeneous Lipschitzian ) LinearOperator of X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) ) holds modetrans (f : ( ( Function-like quasi_total additive homogeneous Lipschitzian ) ( non empty Relation-like the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total additive homogeneous Lipschitzian ) LinearOperator of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) ,b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) ) ,X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) ) : ( ( Function-like quasi_total additive homogeneous Lipschitzian ) ( non empty Relation-like the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total additive homogeneous Lipschitzian ) LinearOperator of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) ,b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) ) = f : ( ( Function-like quasi_total additive homogeneous Lipschitzian ) ( non empty Relation-like the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total additive homogeneous Lipschitzian ) LinearOperator of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) ,b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) ) ;

theorem :: LOPBAN_1:30
for X, Y being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace)
for f being ( ( Function-like quasi_total additive homogeneous Lipschitzian ) ( non empty Relation-like the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total additive homogeneous Lipschitzian ) LinearOperator of X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) ) holds (BoundedLinearOperatorsNorm (X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) )) : ( ( Function-like quasi_total ) ( non empty Relation-like BoundedLinearOperators (b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) ,b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) ) : ( ( ) ( non empty ) Subset of ) -defined REAL : ( ( ) ( non empty V34() V144() V145() V146() V150() ) set ) -valued Function-like total quasi_total V134() V135() V136() ) Function of BoundedLinearOperators (b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) ,b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) ) : ( ( ) ( non empty ) Subset of ) , REAL : ( ( ) ( non empty V34() V144() V145() V146() V150() ) set ) ) . f : ( ( Function-like quasi_total additive homogeneous Lipschitzian ) ( non empty Relation-like the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total additive homogeneous Lipschitzian ) LinearOperator of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) ,b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) ) : ( ( ) ( ext-real V31() real ) Element of REAL : ( ( ) ( non empty V34() V144() V145() V146() V150() ) set ) ) = upper_bound (PreNorms f : ( ( Function-like quasi_total additive homogeneous Lipschitzian ) ( non empty Relation-like the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total additive homogeneous Lipschitzian ) LinearOperator of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) ,b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) ) ) : ( ( non empty ) ( non empty V144() V145() V146() ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( ext-real V31() real ) Element of REAL : ( ( ) ( non empty V34() V144() V145() V146() V150() ) set ) ) ;

definition
let X, Y be ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) ;
func R_NormSpace_of_BoundedLinearOperators (X,Y) -> ( ( non empty ) ( non empty ) NORMSTR ) equals :: LOPBAN_1:def 14
 NORMSTR(# (BoundedLinearOperators (X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) )) : ( ( ) ( non empty ) Subset of ) ,(Zero_ ((BoundedLinearOperators (X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) )) : ( ( ) ( non empty ) Subset of ) ,(R_VectorSpace_of_LinearOperators (X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) )) : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() constituted-Functions ) RealLinearSpace) )) : ( ( ) ( ) Element of BoundedLinearOperators (X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( non empty ) Subset of ) ) ,(Add_ ((BoundedLinearOperators (X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) )) : ( ( ) ( non empty ) Subset of ) ,(R_VectorSpace_of_LinearOperators (X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) )) : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() constituted-Functions ) RealLinearSpace) )) : ( ( Function-like quasi_total ) ( non empty Relation-like [:(BoundedLinearOperators (X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) )) : ( ( ) ( non empty ) Subset of ) ,(BoundedLinearOperators (X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) )) : ( ( ) ( non empty ) Subset of ) :] : ( ( ) ( non empty ) set ) -defined BoundedLinearOperators (X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( non empty ) Subset of ) -valued Function-like total quasi_total ) Element of bool [:[:(BoundedLinearOperators (X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) )) : ( ( ) ( non empty ) Subset of ) ,(BoundedLinearOperators (X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) )) : ( ( ) ( non empty ) Subset of ) :] : ( ( ) ( non empty ) set ) ,(BoundedLinearOperators (X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) )) : ( ( ) ( non empty ) Subset of ) :] : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,(Mult_ ((BoundedLinearOperators (X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) )) : ( ( ) ( non empty ) Subset of ) ,(R_VectorSpace_of_LinearOperators (X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) )) : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ) ( non empty left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() constituted-Functions ) RealLinearSpace) )) : ( ( Function-like quasi_total ) ( non empty Relation-like [:REAL : ( ( ) ( non empty V34() V144() V145() V146() V150() ) set ) ,(BoundedLinearOperators (X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) )) : ( ( ) ( non empty ) Subset of ) :] : ( ( ) ( non empty ) set ) -defined BoundedLinearOperators (X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( non empty ) Subset of ) -valued Function-like total quasi_total ) Element of bool [:[:REAL : ( ( ) ( non empty V34() V144() V145() V146() V150() ) set ) ,(BoundedLinearOperators (X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) )) : ( ( ) ( non empty ) Subset of ) :] : ( ( ) ( non empty ) set ) ,(BoundedLinearOperators (X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) )) : ( ( ) ( non empty ) Subset of ) :] : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,(BoundedLinearOperatorsNorm (X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like BoundedLinearOperators (X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( non empty ) Subset of ) -defined REAL : ( ( ) ( non empty V34() V144() V145() V146() V150() ) set ) -valued Function-like total quasi_total V134() V135() V136() ) Function of BoundedLinearOperators (X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( non empty ) Subset of ) , REAL : ( ( ) ( non empty V34() V144() V145() V146() V150() ) set ) ) #) : ( ( strict ) ( non empty strict ) NORMSTR ) ;
end;

theorem :: LOPBAN_1:31
for X, Y being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) holds the carrier of X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) --> (0. Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) ) : ( ( ) ( V53(b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) ) ) Element of the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like constant total quasi_total ) Element of bool [: the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) = 0. (R_NormSpace_of_BoundedLinearOperators (X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) )) : ( ( non empty ) ( non empty ) NORMSTR ) : ( ( ) ( V53( R_NormSpace_of_BoundedLinearOperators (b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) ,b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) ) : ( ( non empty ) ( non empty ) NORMSTR ) ) ) Element of the carrier of (R_NormSpace_of_BoundedLinearOperators (b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) ,b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) )) : ( ( non empty ) ( non empty ) NORMSTR ) : ( ( ) ( non empty ) set ) ) ;

theorem :: LOPBAN_1:32
for X, Y being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace)
for f being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) )
for g being ( ( Function-like quasi_total additive homogeneous Lipschitzian ) ( non empty Relation-like the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total additive homogeneous Lipschitzian ) LinearOperator of X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) ) st g : ( ( Function-like quasi_total additive homogeneous Lipschitzian ) ( non empty Relation-like the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total additive homogeneous Lipschitzian ) LinearOperator of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) ,b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) ) = f : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) holds
for t being ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) holds ||.(g : ( ( Function-like quasi_total additive homogeneous Lipschitzian ) ( non empty Relation-like the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total additive homogeneous Lipschitzian ) LinearOperator of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) ,b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) ) . t : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) .|| : ( ( ) ( ext-real non negative V31() real ) Element of REAL : ( ( ) ( non empty V34() V144() V145() V146() V150() ) set ) ) <= ||.f : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) .|| : ( ( ) ( ext-real V31() real ) Element of REAL : ( ( ) ( non empty V34() V144() V145() V146() V150() ) set ) ) * ||.t : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) .|| : ( ( ) ( ext-real non negative V31() real ) Element of REAL : ( ( ) ( non empty V34() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ext-real V31() real ) Element of REAL : ( ( ) ( non empty V34() V144() V145() V146() V150() ) set ) ) ;

theorem :: LOPBAN_1:33
for X, Y being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace)
for f being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) holds 0 : ( ( ) ( empty Function-like functional ext-real non positive non negative epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V31() real V113() V114() V144() V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V144() V145() V146() V147() V148() V149() V150() ) Element of bool REAL : ( ( ) ( non empty V34() V144() V145() V146() V150() ) set ) : ( ( ) ( non empty ) set ) ) ) <= ||.f : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) .|| : ( ( ) ( ext-real V31() real ) Element of REAL : ( ( ) ( non empty V34() V144() V145() V146() V150() ) set ) ) ;

theorem :: LOPBAN_1:34
for X, Y being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace)
for f being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) st f : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) = 0. (R_NormSpace_of_BoundedLinearOperators (X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) )) : ( ( non empty ) ( non empty ) NORMSTR ) : ( ( ) ( V53( R_NormSpace_of_BoundedLinearOperators (b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) ,b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) ) : ( ( non empty ) ( non empty ) NORMSTR ) ) ) Element of the carrier of (R_NormSpace_of_BoundedLinearOperators (b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) ,b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) )) : ( ( non empty ) ( non empty ) NORMSTR ) : ( ( ) ( non empty ) set ) ) holds
0 : ( ( ) ( empty Function-like functional ext-real non positive non negative epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V31() real V113() V114() V144() V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V144() V145() V146() V147() V148() V149() V150() ) Element of bool REAL : ( ( ) ( non empty V34() V144() V145() V146() V150() ) set ) : ( ( ) ( non empty ) set ) ) ) = ||.f : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) .|| : ( ( ) ( ext-real V31() real ) Element of REAL : ( ( ) ( non empty V34() V144() V145() V146() V150() ) set ) ) ;

registration
let X, Y be ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) ;
cluster -> Relation-like Function-like for ( ( ) ( ) Element of the carrier of (R_NormSpace_of_BoundedLinearOperators (X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) )) : ( ( non empty ) ( non empty ) NORMSTR ) : ( ( ) ( non empty ) set ) ) ;
end;

definition
let X, Y be ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) ;
let f be ( ( ) ( Relation-like Function-like ) Element of ( ( ) ( non empty ) set ) ) ;
let v be ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) ;
:: original: .
redefine func f . v -> ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) ;
end;

theorem :: LOPBAN_1:35
for X, Y being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace)
for f, g, h being ( ( ) ( Relation-like Function-like ) Point of ( ( ) ( non empty ) set ) ) holds
( h : ( ( ) ( Relation-like Function-like ) Point of ( ( ) ( non empty ) set ) ) = f : ( ( ) ( Relation-like Function-like ) Point of ( ( ) ( non empty ) set ) ) + g : ( ( ) ( Relation-like Function-like ) Point of ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like Function-like ) Element of the carrier of (R_NormSpace_of_BoundedLinearOperators (b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) ,b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) )) : ( ( non empty ) ( non empty ) NORMSTR ) : ( ( ) ( non empty ) set ) ) iff for x being ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) holds h : ( ( ) ( Relation-like Function-like ) Point of ( ( ) ( non empty ) set ) ) . x : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) = (f : ( ( ) ( Relation-like Function-like ) Point of ( ( ) ( non empty ) set ) ) . x : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) + (g : ( ( ) ( Relation-like Function-like ) Point of ( ( ) ( non empty ) set ) ) . x : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) ) ;

theorem :: LOPBAN_1:36
for X, Y being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace)
for f, h being ( ( ) ( Relation-like Function-like ) Point of ( ( ) ( non empty ) set ) )
for a being ( ( ) ( ext-real V31() real ) Real) holds
( h : ( ( ) ( Relation-like Function-like ) Point of ( ( ) ( non empty ) set ) ) = a : ( ( ) ( ext-real V31() real ) Real) * f : ( ( ) ( Relation-like Function-like ) Point of ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like Function-like ) Element of the carrier of (R_NormSpace_of_BoundedLinearOperators (b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) ,b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) )) : ( ( non empty ) ( non empty ) NORMSTR ) : ( ( ) ( non empty ) set ) ) iff for x being ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) holds h : ( ( ) ( Relation-like Function-like ) Point of ( ( ) ( non empty ) set ) ) . x : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) = a : ( ( ) ( ext-real V31() real ) Real) * (f : ( ( ) ( Relation-like Function-like ) Point of ( ( ) ( non empty ) set ) ) . x : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) ) ;

theorem :: LOPBAN_1:37
for X, Y being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace)
for f, g being ( ( ) ( Relation-like Function-like ) Point of ( ( ) ( non empty ) set ) )
for a being ( ( ) ( ext-real V31() real ) Real) holds
( ( ||.f : ( ( ) ( Relation-like Function-like ) Point of ( ( ) ( non empty ) set ) ) .|| : ( ( ) ( ext-real V31() real ) Element of REAL : ( ( ) ( non empty V34() V144() V145() V146() V150() ) set ) ) = 0 : ( ( ) ( empty Function-like functional ext-real non positive non negative epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V31() real V113() V114() V144() V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V144() V145() V146() V147() V148() V149() V150() ) Element of bool REAL : ( ( ) ( non empty V34() V144() V145() V146() V150() ) set ) : ( ( ) ( non empty ) set ) ) ) implies f : ( ( ) ( Relation-like Function-like ) Point of ( ( ) ( non empty ) set ) ) = 0. (R_NormSpace_of_BoundedLinearOperators (X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) )) : ( ( non empty ) ( non empty ) NORMSTR ) : ( ( ) ( Relation-like Function-like V53( R_NormSpace_of_BoundedLinearOperators (b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) ,b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) ) : ( ( non empty ) ( non empty ) NORMSTR ) ) ) Element of the carrier of (R_NormSpace_of_BoundedLinearOperators (b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) ,b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) )) : ( ( non empty ) ( non empty ) NORMSTR ) : ( ( ) ( non empty ) set ) ) ) & ( f : ( ( ) ( Relation-like Function-like ) Point of ( ( ) ( non empty ) set ) ) = 0. (R_NormSpace_of_BoundedLinearOperators (X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) )) : ( ( non empty ) ( non empty ) NORMSTR ) : ( ( ) ( Relation-like Function-like V53( R_NormSpace_of_BoundedLinearOperators (b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) ,b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) ) : ( ( non empty ) ( non empty ) NORMSTR ) ) ) Element of the carrier of (R_NormSpace_of_BoundedLinearOperators (b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) ,b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) )) : ( ( non empty ) ( non empty ) NORMSTR ) : ( ( ) ( non empty ) set ) ) implies ||.f : ( ( ) ( Relation-like Function-like ) Point of ( ( ) ( non empty ) set ) ) .|| : ( ( ) ( ext-real V31() real ) Element of REAL : ( ( ) ( non empty V34() V144() V145() V146() V150() ) set ) ) = 0 : ( ( ) ( empty Function-like functional ext-real non positive non negative epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V31() real V113() V114() V144() V145() V146() V147() V148() V149() V150() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V144() V145() V146() V147() V148() V149() V150() ) Element of bool REAL : ( ( ) ( non empty V34() V144() V145() V146() V150() ) set ) : ( ( ) ( non empty ) set ) ) ) ) & ||.(a : ( ( ) ( ext-real V31() real ) Real) * f : ( ( ) ( Relation-like Function-like ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( Relation-like Function-like ) Element of the carrier of (R_NormSpace_of_BoundedLinearOperators (b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) ,b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) )) : ( ( non empty ) ( non empty ) NORMSTR ) : ( ( ) ( non empty ) set ) ) .|| : ( ( ) ( ext-real V31() real ) Element of REAL : ( ( ) ( non empty V34() V144() V145() V146() V150() ) set ) ) = (abs a : ( ( ) ( ext-real V31() real ) Real) ) : ( ( ) ( ext-real V31() real ) Element of REAL : ( ( ) ( non empty V34() V144() V145() V146() V150() ) set ) ) * ||.f : ( ( ) ( Relation-like Function-like ) Point of ( ( ) ( non empty ) set ) ) .|| : ( ( ) ( ext-real V31() real ) Element of REAL : ( ( ) ( non empty V34() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ext-real V31() real ) Element of REAL : ( ( ) ( non empty V34() V144() V145() V146() V150() ) set ) ) & ||.(f : ( ( ) ( Relation-like Function-like ) Point of ( ( ) ( non empty ) set ) ) + g : ( ( ) ( Relation-like Function-like ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( Relation-like Function-like ) Element of the carrier of (R_NormSpace_of_BoundedLinearOperators (b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) ,b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) )) : ( ( non empty ) ( non empty ) NORMSTR ) : ( ( ) ( non empty ) set ) ) .|| : ( ( ) ( ext-real V31() real ) Element of REAL : ( ( ) ( non empty V34() V144() V145() V146() V150() ) set ) ) <= ||.f : ( ( ) ( Relation-like Function-like ) Point of ( ( ) ( non empty ) set ) ) .|| : ( ( ) ( ext-real V31() real ) Element of REAL : ( ( ) ( non empty V34() V144() V145() V146() V150() ) set ) ) + ||.g : ( ( ) ( Relation-like Function-like ) Point of ( ( ) ( non empty ) set ) ) .|| : ( ( ) ( ext-real V31() real ) Element of REAL : ( ( ) ( non empty V34() V144() V145() V146() V150() ) set ) ) : ( ( ) ( ext-real V31() real ) Element of REAL : ( ( ) ( non empty V34() V144() V145() V146() V150() ) set ) ) ) ;

theorem :: LOPBAN_1:38
for X, Y being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) holds
( R_NormSpace_of_BoundedLinearOperators (X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) ) : ( ( non empty ) ( non empty ) NORMSTR ) is reflexive & R_NormSpace_of_BoundedLinearOperators (X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) ) : ( ( non empty ) ( non empty ) NORMSTR ) is discerning & R_NormSpace_of_BoundedLinearOperators (X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) ) : ( ( non empty ) ( non empty ) NORMSTR ) is RealNormSpace-like ) ;

theorem :: LOPBAN_1:39
for X, Y being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) holds R_NormSpace_of_BoundedLinearOperators (X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) ) : ( ( non empty ) ( non empty ) NORMSTR ) is ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) ;

registration
let X, Y be ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) ;
cluster R_NormSpace_of_BoundedLinearOperators (X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) ) : ( ( non empty ) ( non empty ) NORMSTR ) -> non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ;
end;

theorem :: LOPBAN_1:40
for X, Y being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace)
for f, g, h being ( ( ) ( Relation-like Function-like ) Point of ( ( ) ( non empty ) set ) ) holds
( h : ( ( ) ( Relation-like Function-like ) Point of ( ( ) ( non empty ) set ) ) = f : ( ( ) ( Relation-like Function-like ) Point of ( ( ) ( non empty ) set ) ) - g : ( ( ) ( Relation-like Function-like ) Point of ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like Function-like ) Element of the carrier of (R_NormSpace_of_BoundedLinearOperators (b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) ,b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) )) : ( ( non empty ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) : ( ( ) ( non empty ) set ) ) iff for x being ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) holds h : ( ( ) ( Relation-like Function-like ) Point of ( ( ) ( non empty ) set ) ) . x : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) = (f : ( ( ) ( Relation-like Function-like ) Point of ( ( ) ( non empty ) set ) ) . x : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) - (g : ( ( ) ( Relation-like Function-like ) Point of ( ( ) ( non empty ) set ) ) . x : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) VECTOR of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) ) ;

begin

definition
let X be ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) ;
attr X is complete means :: LOPBAN_1:def 15
 for seq being ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V144() V145() V146() V147() V148() V149() V150() ) Element of bool REAL : ( ( ) ( non empty V34() V144() V145() V146() V150() ) set ) : ( ( ) ( non empty ) set ) ) -defined the carrier of X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) st seq : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V144() V145() V146() V147() V148() V149() V150() ) Element of bool REAL : ( ( ) ( non empty V34() V144() V145() V146() V150() ) set ) : ( ( ) ( non empty ) set ) ) -defined the carrier of X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) is Cauchy_sequence_by_Norm holds
seq : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V144() V145() V146() V147() V148() V149() V150() ) Element of bool REAL : ( ( ) ( non empty V34() V144() V145() V146() V150() ) set ) : ( ( ) ( non empty ) set ) ) -defined the carrier of X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) is convergent ;
end;

registration
cluster non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like complete for ( ( ) ( ) NORMSTR ) ;
end;

definition
mode RealBanachSpace is ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like complete ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like complete ) RealNormSpace) ;
end;

theorem :: LOPBAN_1:41
for X being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace)
for seq being ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V144() V145() V146() V147() V148() V149() V150() ) Element of bool REAL : ( ( ) ( non empty V34() V144() V145() V146() V150() ) set ) : ( ( ) ( non empty ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) st seq : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V144() V145() V146() V147() V148() V149() V150() ) Element of bool REAL : ( ( ) ( non empty V34() V144() V145() V146() V150() ) set ) : ( ( ) ( non empty ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) is convergent holds
( ||.seq : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V144() V145() V146() V147() V148() V149() V150() ) Element of bool REAL : ( ( ) ( non empty V34() V144() V145() V146() V150() ) set ) : ( ( ) ( non empty ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) .|| : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V144() V145() V146() V147() V148() V149() V150() ) Element of bool REAL : ( ( ) ( non empty V34() V144() V145() V146() V150() ) set ) : ( ( ) ( non empty ) set ) ) -defined REAL : ( ( ) ( non empty V34() V144() V145() V146() V150() ) set ) -valued Function-like total quasi_total V134() V135() V136() ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V144() V145() V146() V147() V148() V149() V150() ) Element of bool REAL : ( ( ) ( non empty V34() V144() V145() V146() V150() ) set ) : ( ( ) ( non empty ) set ) ) ,REAL : ( ( ) ( non empty V34() V144() V145() V146() V150() ) set ) :] : ( ( ) ( non empty V134() V135() V136() ) set ) : ( ( ) ( non empty ) set ) ) is convergent & lim ||.seq : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V144() V145() V146() V147() V148() V149() V150() ) Element of bool REAL : ( ( ) ( non empty V34() V144() V145() V146() V150() ) set ) : ( ( ) ( non empty ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) .|| : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V144() V145() V146() V147() V148() V149() V150() ) Element of bool REAL : ( ( ) ( non empty V34() V144() V145() V146() V150() ) set ) : ( ( ) ( non empty ) set ) ) -defined REAL : ( ( ) ( non empty V34() V144() V145() V146() V150() ) set ) -valued Function-like total quasi_total V134() V135() V136() ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V144() V145() V146() V147() V148() V149() V150() ) Element of bool REAL : ( ( ) ( non empty V34() V144() V145() V146() V150() ) set ) : ( ( ) ( non empty ) set ) ) ,REAL : ( ( ) ( non empty V34() V144() V145() V146() V150() ) set ) :] : ( ( ) ( non empty V134() V135() V136() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ext-real V31() real ) Element of REAL : ( ( ) ( non empty V34() V144() V145() V146() V150() ) set ) ) = ||.(lim seq : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V144() V145() V146() V147() V148() V149() V150() ) Element of bool REAL : ( ( ) ( non empty V34() V144() V145() V146() V150() ) set ) : ( ( ) ( non empty ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) : ( ( ) ( non empty ) set ) ) .|| : ( ( ) ( ext-real non negative V31() real ) Element of REAL : ( ( ) ( non empty V34() V144() V145() V146() V150() ) set ) ) ) ;

theorem :: LOPBAN_1:42
for X, Y being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) st Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) is complete holds
for seq being ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V144() V145() V146() V147() V148() V149() V150() ) Element of bool REAL : ( ( ) ( non empty V34() V144() V145() V146() V150() ) set ) : ( ( ) ( non empty ) set ) ) -defined the carrier of (R_NormSpace_of_BoundedLinearOperators (b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) ,b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) )) : ( ( non empty ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) st seq : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V144() V145() V146() V147() V148() V149() V150() ) Element of bool REAL : ( ( ) ( non empty V34() V144() V145() V146() V150() ) set ) : ( ( ) ( non empty ) set ) ) -defined the carrier of (R_NormSpace_of_BoundedLinearOperators (b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) ,b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) )) : ( ( non empty ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) is Cauchy_sequence_by_Norm holds
seq : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V144() V145() V146() V147() V148() V149() V150() ) Element of bool REAL : ( ( ) ( non empty V34() V144() V145() V146() V150() ) set ) : ( ( ) ( non empty ) set ) ) -defined the carrier of (R_NormSpace_of_BoundedLinearOperators (b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) ,b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) )) : ( ( non empty ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) sequence of ( ( ) ( non empty ) set ) ) is convergent ;

theorem :: LOPBAN_1:43
for X being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace)
for Y being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like complete ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like complete ) RealBanachSpace) holds R_NormSpace_of_BoundedLinearOperators (X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like complete ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like complete ) RealBanachSpace) ) : ( ( non empty ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) is ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like complete ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like complete ) RealBanachSpace) ;

registration
let X be ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) RealNormSpace) ;
let Y be ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like complete ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like complete ) RealBanachSpace) ;
cluster R_NormSpace_of_BoundedLinearOperators (X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) ,Y : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital discerning reflexive RealNormSpace-like complete ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like complete ) NORMSTR ) ) : ( ( non empty ) ( non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V104() discerning reflexive RealNormSpace-like ) NORMSTR ) -> non empty complete ;
end;